Seasonal affective disorder

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Seasonal_affective_disorder 

Seasonal affective disorder
Other names	Seasonal mood disorder, depressive disorder with seasonal pattern, winter depression, winter blues, summer depression, seasonal depression
Bright light therapy is a common treatment for seasonal affective disorder and for circadian rhythm sleep disorders.
Specialty	Psychiatry

Seasonal affective disorder (SAD) is a mood disorder subset in which people who have normal mental health throughout most of the year exhibit depressive symptoms at the same time each year, most commonly in winter. Common symptoms include sleeping too much, having little to no energy, and overeating. The condition in the summer can include heightened anxiety.

In the Diagnostic and Statistical Manual of Mental Disorders DSM-IV and DSM-5, its status was changed. It is no longer classified as a unique mood disorder but is now a specifier, called "with seasonal pattern", for recurrent major depressive disorder that occurs at a specific time of the year and fully remits otherwise. Although experts were initially skeptical, this condition is now recognized as a common disorder. However, the validity of SAD has been questioned by a 2016 analysis by the Center for Disease Control, in which no links were detected between depression and seasonality or sunlight exposure.

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Video explanation

In the United States, the percentage of the population affected by SAD ranges from 1.4% of the population in Florida, to 9.9% in Alaska. SAD was formally described and named in 1984 by Norman E. Rosenthal and colleagues at the National Institute of Mental Health.

History

SAD was first systematically reported and named in the early 1980s by Norman E. Rosenthal, M.D., and his associates at the National Institute of Mental Health (NIMH). Rosenthal was initially motivated by his desire to discover the cause of his own experience of depression during the dark days of the northern US winter, called polar night. He theorized that the reduction in available natural light during winter was the cause. Rosenthal and his colleagues then documented the phenomenon of SAD in a placebo-controlled study utilizing light therapy. A paper based on this research was published in 1984. Although Rosenthal's ideas were initially greeted with skepticism, SAD has become well recognized, and his 1993 book, Winter Blues has become the standard introduction to the subject.

Research on SAD in the United States began in 1979 when Herb Kern, a research engineer, had also noticed that he felt depressed during the winter months. Kern suspected that scarcer light in winter was the cause and discussed the idea with scientists at the NIMH who were working on bodily rhythms. They were intrigued, and responded by devising a lightbox to treat Kern’s depression. Kern felt much better within a few days of treatments, as did other patients treated in the same way.

Signs and symptoms

SAD is a type of major depressive disorder, and sufferers may exhibit any of the associated symptoms, such as feelings of hopelessness and worthlessness, thoughts of suicide, loss of interest in activities, withdrawal from social interaction, sleep and appetite problems, difficulty with concentrating and making decisions, decreased libido, a lack of energy, or agitation. Symptoms of winter SAD often include falling asleep earlier or in less than 5 minutes in the evening, oversleeping or difficulty waking up in the morning, nausea, and a tendency to overeat, often with a craving for carbohydrates, which leads to weight gain. SAD is typically associated with winter depression, but springtime lethargy or other seasonal mood patterns are not uncommon. Although each individual case is different, in contrast to winter SAD, people who experience spring and summer depression may be more likely to show symptoms such as insomnia, decreased appetite and weight loss, and agitation or anxiety.

Bipolar disorder

With seasonal pattern is a specifier for bipolar and related disorders, including bipolar I disorder and bipolar II disorder. Most people with SAD experience major depressive disorder, but as many as 20% may have a bipolar disorder. It is important to discriminate between diagnoses because there are important treatment differences. In these cases, people who have the With seasonal pattern specifier may experience a depressive episode either due to major depressive disorder or as part of bipolar disorder during the winter and remit in the summer. Around 25% of patients with bipolar disorder may present with a depressive seasonal pattern, which is associated with bipolar II disorder, rapid cycling, eating disorders, and more depressive episodes. Differences in biological sex display distinct clinical characteristics associated to seasonal pattern: males present with more Bipolar II disorder and a higher number of depressive episodes, and females with rapid cycling and eating disorders.

Cause

In many species, activity is diminished during the winter months in response to the reduction in available food, the reduction of sunlight (especially for diurnal animals) and the difficulties of surviving in cold weather. Hibernation is an extreme example, but even species that do not hibernate often exhibit changes in behavior during the winter. The preponderance of women with SAD suggests that the response may also somehow regulate reproduction.

Various proximate causes have been proposed. One possibility is that SAD is related to a lack of serotonin, and serotonin polymorphisms could play a role in SAD, although this has been disputed. Mice incapable of turning serotonin into N-acetylserotonin (by serotonin N-acetyltransferase) appear to express "depression-like" behavior, and antidepressants such as fluoxetine increase the amount of the enzyme serotonin N-acetyltransferase, resulting in an antidepressant-like effect. Another theory is that the cause may be related to melatonin which is produced in dim light and darkness by the pineal gland, since there are direct connections, via the retinohypothalamic tract and the suprachiasmatic nucleus, between the retina and the pineal gland. Melatonin secretion is controlled by the endogenous circadian clock, but can also be suppressed by bright light.

One study looked at whether some people could be predisposed to SAD based on personality traits. Correlations between certain personality traits, higher levels of neuroticism, agreeableness, openness, and an avoidance-oriented coping style, appeared to be common in those with SAD.

Pathophysiology

Seasonal mood variations are believed to be related to light. An argument for this view is the effectiveness of bright-light therapy. SAD is measurably present at latitudes in the Arctic region, such as northern Finland (64°00′N), where the rate of SAD is 9.5%. Cloud cover may contribute to the negative effects of SAD. There is evidence that many patients with SAD have a delay in their circadian rhythm, and that bright light treatment corrects these delays which may be responsible for the improvement in patients.

The symptoms of it mimic those of dysthymia or even major depressive disorder. There is also potential risk of suicide in some patients experiencing SAD. One study reports 6–35% of sufferers required hospitalization during one period of illness. At times, patients may not feel depressed, but rather lack energy to perform everyday activities.

Subsyndromal Seasonal Affective Disorder is a milder form of SAD experienced by an estimated 14.3% (vs. 6.1% SAD) of the U.S. population. The blue feeling experienced by both SAD and SSAD sufferers can usually be dampened or extinguished by exercise and increased outdoor activity, particularly on sunny days, resulting in increased solar exposure. Connections between human mood, as well as energy levels, and the seasons are well documented, even in healthy individuals.

Diagnosis

According to the American Psychiatric Association DSM-IV criteria, Seasonal Affective Disorder is not regarded as a separate disorder. It is called a "course specifier" and may be applied as an added description to the pattern of major depressive episodes in patients with major depressive disorder or patients with bipolar disorder.

The "Seasonal Pattern Specifier" must meet four criteria: depressive episodes at a particular time of the year; remissions or mania/hypomania at a characteristic time of year; these patterns must have lasted two years with no nonseasonal major depressive episodes during that same period; and these seasonal depressive episodes outnumber other depressive episodes throughout the patient's lifetime. The Mayo Clinic describes three types of SAD, each with its own set of symptoms.

Management

See also: Occupational therapy in the management of seasonal affective disorder

Treatments for classic (winter-based) seasonal affective disorder include light therapy, medication, ionized-air administration, cognitive-behavioral therapy and carefully timed supplementation of the hormone melatonin.

Light therapy

Photoperiod-related alterations of the duration of melatonin secretion may affect the seasonal mood cycles of SAD. This suggests that light therapy may be an effective treatment for SAD. Light therapy uses a lightbox which emits far more lumens than a customary incandescent lamp. Bright white "full spectrum" light at 10,000 lux, blue light at a wavelength of 480 nm at 2,500 lux or green (actually cyan or blue-green) light at a wavelength of 500 nm at 350 lux are used, with the first-mentioned historically preferred.

Bright light therapy is effective with the patient sitting a prescribed distance, commonly 30–60 cm, in front of the box with her/his eyes open but not staring at the light source for 30–60 minutes. A study published in May 2010 suggests that the blue light often used for SAD treatment should perhaps be replaced by green or white illumination. Discovering the best schedule is essential. One study has shown that up to 69% of patients find lightbox treatment inconvenient and as many as 19% stop use because of this.

Dawn simulation has also proven to be effective; in some studies, there is an 83% better response when compared to other bright light therapy. When compared in a study to negative air ionization, bright light was shown to be 57% effective vs. dawn simulation 50%. Patients using light therapy can experience improvement during the first week, but increased results are evident when continued throughout several weeks. Certain symptoms like hypersomnia, early insomnia, social withdrawal, and anxiety resolve more rapidly with light therapy than with cognitive behavioral therapy. Most studies have found it effective without use year round but rather as a seasonal treatment lasting for several weeks until frequent light exposure is naturally obtained.

Light therapy can also consist of exposure to sunlight, either by spending more time outside or using a computer-controlled heliostat to reflect sunlight into the windows of a home or office. Although light therapy is the leading treatment for seasonal affective disorder, prolonged direct sunlight or artificial lights that don't block the ultraviolet range should be avoided due to the threat of skin cancer.

The evidence base for light therapy as a preventive treatment for seasonal affective disorder is limited. The decision to use light therapy to treat people with a history of winter depression before depressive symptoms begin should be based on a persons preference of treatment.

Medication

SSRI (selective serotonin reuptake inhibitor) antidepressants have proven effective in treating SAD. Effective antidepressants are fluoxetine, sertraline, or paroxetine. Both fluoxetine and light therapy are 67% effective in treating SAD according to direct head-to-head trials conducted during the 2006 Can-SAD study. Subjects using the light therapy protocol showed earlier clinical improvement, generally within one week of beginning the clinical treatment. Bupropion extended-release has been shown to prevent SAD for one in four people, but has not been compared directly to other preventive options in trials. In a 2021 updated Cochrane review of second-generation antidepressant medications for the treatment of SAD a definitive conclusion could not be drawn due to lack of evidence and the need for larger randomized controlled trials.

Modafinil may be an effective and well-tolerated treatment in patients with seasonal affective disorder/winter depression.

Another explanation is that vitamin D levels are too low when people do not get enough Ultraviolet-B on their skin. An alternative to using bright lights is to take vitamin D supplements. However, studies did not show a link between vitamin D levels and depressive symptoms in elderly Chinese nor among elderly British women given only 800IU when 6,000IU is needed. 5-HTP (an amino acid that helps to produce serotonin and is often used to help those with depression) has also been suggested as a supplement that may help treat the symptoms of SAD, by lifting mood and regulating sleep schedule for sufferers. However, those who take antidepressants are not advised to take 5-HTP, as antidepressant medications may combine with the supplement to create dangerously high levels of serotonin – potentially resulting in 'serotonin syndrome'.

Other treatments

Depending upon the patient, one treatment (e.g., lightbox) may be used in conjunction with another (e.g., medication).

Negative air ionization, which involves releasing charged particles into the sleep environment, has been found effective with a 47.9% improvement if the negative ions are in sufficient density (quantity).

Physical exercise has shown to be an effective form of depression therapy, particularly when in addition to another form of treatment for SAD. One particular study noted marked effectiveness for treatment of depressive symptoms when combining regular exercise with bright light therapy. Patients exposed to exercise which had been added to their treatments in 20 minutes intervals on the aerobic bike during the day along with the same amount of time underneath the UV light were seen to make quick recovery.

Of all the psychological therapies aimed at the prevention of SAD, cognitive-behaviour therapy, typically involving thought records, activity schedules and a positive data log, has been the subject of the most empirical work, however, evidence for CBT or any of the psychological therapies aimed at preventing SAD remains inconclusive.

Epidemiology

Nordic countries

Winter depression is a common slump in the mood of some inhabitants of most of the Nordic countries. Iceland, however, seems to be an exception. A study of more than 2000 people there found the prevalence of seasonal affective disorder and seasonal changes in anxiety and depression to be unexpectedly low in both sexes. The study's authors suggested that propensity for SAD may differ due to some genetic factor within the Icelandic population. A study of Canadians of wholly Icelandic descent also showed low levels of SAD. It has more recently been suggested that this may be attributed to the large amount of fish traditionally eaten by Icelandic people, in 2007 about 90 kilograms per person per year as opposed to about 24 kg in the US and Canada, rather than to genetic predisposition; a similar anomaly is noted in Japan, where annual fish consumption in recent years averages about 60 kg per capita. Fish are high in vitamin D. Fish also contain docosahexaenoic acid (DHA), which help with a variety of neurological dysfunctions.

Other countries

In the United States, a diagnosis of seasonal affective disorder was first proposed by Norman E. Rosenthal, M.D. in 1984. Rosenthal wondered why he became sluggish during the winter after moving from sunny South Africa to (cloudy in winter) New York. He started experimenting increasing exposure to artificial light, and found this made a difference. In Alaska it has been established that there is a SAD rate of 8.9%, and an even greater rate of 24.9% for subsyndromal SAD.

Around 20% of Irish people are affected by SAD, according to a survey conducted in 2007. The survey also shows women are more likely to be affected by SAD than men. An estimated 3% of the population in the Netherlands suffer from winter SAD.

Integration by parts

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Integration_by_parts 

 

Part of a series of articles about
Calculus
Fundamental theorem Leibniz integral rule Limits of functions Continuity Mean value theorem Rolle's theorem

Differential

Integral Lists of integrals Integral transform Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, Weierstrass, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.

The integration by parts formula states:

$${\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}$$

Or, letting ${\displaystyle u=u(x)}$ and ${\displaystyle du=u'(x)\,dx}$ while ${\displaystyle v=v(x)}$ and ${\displaystyle dv=v'(x)\,dx}$, the formula can be written more compactly:

$${\displaystyle \int u\,dv\ =\ uv-\int v\,du.}$$

Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.  More general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts.

Theorem

Product of two functions

The theorem can be derived as follows. For two continuously differentiable functions u(x) and v(x), the product rule states:

$${\displaystyle {\Big (}u(x)v(x){\Big )}'=v(x)u'(x)+u(x)v'(x).}$$

Integrating both sides with respect to x,

$${\displaystyle \int {\Big (}u(x)v(x){\Big )}'\,dx=\int u'(x)v(x)\,dx+\int u(x)v'(x)\,dx,}$$

and noting that an indefinite integral is an antiderivative gives

$${\displaystyle u(x)v(x)=\int u'(x)v(x)\,dx+\int u(x)v'(x)\,dx,}$$

where we neglect writing the constant of integration. This yields the formula for integration by parts:

$${\displaystyle \int u(x)v'(x)\,dx=u(x)v(x)-\int u'(x)v(x)\,dx,}$$

or in terms of the differentials ${\displaystyle du=u'(x)\,dx}$, ${\displaystyle dv=v'(x)\,dx,\quad }$

$${\displaystyle \int u(x)\,dv=u(x)v(x)-\int v(x)\,du.}$$

This is to be understood as an equality of functions with an unspecified constant added to each side. Taking the difference of each side between two values x = a and x = b and applying the fundamental theorem of calculus gives the definite integral version:

$${\displaystyle \int _{a}^{b}u(x)v'(x)\,dx=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.}$$

The original integral ∫ uv′ dx contains the derivative v′; to apply the theorem, one must find v, the antiderivative of v', then evaluate the resulting integral ∫ vu′ dx.

Validity for less smooth functions

It is not necessary for u and v to be continuously differentiable. Integration by parts works if u is absolutely continuous and the function designated v′ is Lebesgue integrable (but not necessarily continuous). (If v′ has a point of discontinuity then its antiderivative v may not have a derivative at that point.)

If the interval of integration is not compact, then it is not necessary for u to be absolutely continuous in the whole interval or for v′ to be Lebesgue integrable in the interval, as a couple of examples (in which u and v are continuous and continuously differentiable) will show. For instance, if

$${\displaystyle u(x)=e^{x}/x^{2},\,v'(x)=e^{-x}}$$

u is not absolutely continuous on the interval [1, ∞), but nevertheless

$${\displaystyle \int _{1}^{\infty }u(x)v'(x)\,dx={\Big [}u(x)v(x){\Big ]}_{1}^{\infty }-\int _{1}^{\infty }u'(x)v(x)\,dx}$$

so long as ${\displaystyle \left[u(x)v(x)\right]_{1}^{\infty }}$ is taken to mean the limit of ${\displaystyle u(L)v(L)-u(1)v(1)}$ as ${\displaystyle L\to \infty }$ and so long as the two terms on the right-hand side are finite. This is only true if we choose ${\displaystyle v(x)=-e^{-x}.}$ Similarly, if

$${\displaystyle u(x)=e^{-x},\,v'(x)=x^{-1}\sin(x)}$$

v′ is not Lebesgue integrable on the interval [1, ∞), but nevertheless

$${\displaystyle \int _{1}^{\infty }u(x)v'(x)\,dx={\Big [}u(x)v(x){\Big ]}_{1}^{\infty }-\int _{1}^{\infty }u'(x)v(x)\,dx}$$

with the same interpretation.

One can also easily come up with similar examples in which u and v are not continuously differentiable.

Further, if ${\displaystyle f(x)}$ is a function of bounded variation on the segment ${\displaystyle [a,b],}$ and ${\displaystyle \varphi (x)}$ is differentiable on ${\displaystyle [a,b],}$ then

$${\displaystyle \int _{a}^{b}f(x)\varphi '(x)\,dx=-\int _{-\infty }^{\infty }{\widetilde {\varphi }}(x)\,d({\widetilde {\chi }}_{[a,b]}(x){\widetilde {f}}(x)),}$$

where ${\displaystyle d(\chi _{[a,b]}(x){\widetilde {f}}(x))}$ denotes the signed measure corresponding to the function of bounded variation ${\displaystyle \chi _{[a,b]}(x)f(x)}$, and functions ${\displaystyle {\widetilde {f}},{\widetilde {\varphi }}}$ are extensions of ${\displaystyle f,\varphi }$ to ${\displaystyle \mathbb {R} ,}$ which are respectively of bounded variation and differentiable.

Product of many functions

Integrating the product rule for three multiplied functions, u(x), v(x), w(x), gives a similar result:

$${\displaystyle \int _{a}^{b}uv\,dw\ =\ {\Big [}uvw{\Big ]}_{a}^{b}-\int _{a}^{b}uw\,dv-\int _{a}^{b}vw\,du.}$$

In general, for n factors

$${\displaystyle \left(\prod _{i=1}^{n}u_{i}(x)\right)'\ =\ \sum _{j=1}^{n}u_{j}'(x)\prod _{i\neq j}^{n}u_{i}(x),}$$

which leads to

$${\displaystyle \left[\prod _{i=1}^{n}u_{i}(x)\right]_{a}^{b}\ =\ \sum _{j=1}^{n}\int _{a}^{b}u_{j}'(x)\prod _{i\neq j}^{n}u_{i}(x).}$$

Visualization

Graphical interpretation of the theorem. The pictured curve is parametrized by the variable t.

Consider a parametric curve by (x, y) = (f(t), g(t)). Assuming that the curve is locally one-to-one and integrable, we can define

${\displaystyle x(y)=f(g^{-1}(y))}$

${\displaystyle y(x)=g(f^{-1}(x))}$

The area of the blue region is

${\displaystyle A_{1}=\int _{y_{1}}^{y_{2}}x(y)\,dy}$

Similarly, the area of the red region is

${\displaystyle A_{2}=\int _{x_{1}}^{x_{2}}y(x)\,dx}$

The total area A₁ + A₂ is equal to the area of the bigger rectangle, x₂y₂, minus the area of the smaller one, x₁y₁:

${\displaystyle \overbrace {\int _{y_{1}}^{y_{2}}x(y)\,dy} ^{A_{1}}+\overbrace {\int _{x_{1}}^{x_{2}}y(x)\,dx} ^{A_{2}}\ =\ {\biggl .}x\cdot y(x){\biggl |}_{x_{1}}^{x_{2}}\ =\ {\biggl .}y\cdot x(y){\biggl |}_{y_{1}}^{y_{2}}.}$

Or, in terms of t,

${\displaystyle \int _{t_{1}}^{t_{2}}x(t)\,dy(t)+\int _{t_{1}}^{t_{2}}y(t)\,dx(t)\ =\ {\biggl .}x(t)y(t){\biggl |}_{t_{1}}^{t_{2}}}$

Or, in terms of indefinite integrals, this can be written as

${\displaystyle \int x\,dy+\int y\,dx\ =\ xy}$

Rearranging:

${\displaystyle \int x\,dy\ =\ xy-\int y\,dx}$

Thus integration by parts may be thought of as deriving the area of the blue region from the area of rectangles and that of the red region.

This visualization also explains why integration by parts may help find the integral of an inverse function f⁻¹(x) when the integral of the function f(x) is known. Indeed, the functions x(y) and y(x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx. In particular, this explains use of integration by parts to integrate logarithm and inverse trigonometric functions. In fact, if ${\displaystyle f}$ is a differentiable one-to-one function on an interval, then integration by parts can be used to derive a formula for the integral of ${\displaystyle f^{-1}}$in terms of the integral of ${\displaystyle f}$. This is demonstrated in the article, Integral of inverse functions.

Applications

Finding antiderivatives

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the single function. The following form is useful in illustrating the best strategy to take:

${\displaystyle \int uv\ dx=u\int v\ dx-\int \left(u'\int v\ dx\right)\ dx.}$

On the right-hand side, u is differentiated and v is integrated; consequently it is useful to choose u as a function that simplifies when differentiated, or to choose v as a function that simplifies when integrated. As a simple example, consider:

${\displaystyle \int {\frac {\ln(x)}{x^{2}}}\ dx\ .}$

Since the derivative of ln(x) is 1/x, one makes (ln(x)) part u; since the antiderivative of 1/x² is −1/x, one makes 1/x² dx part dv. The formula now yields:

${\displaystyle \int {\frac {\ln(x)}{x^{2}}}\ dx=-{\frac {\ln(x)}{x}}-\int {\biggl (}{\frac {1}{x}}{\biggr )}{\biggl (}-{\frac {1}{x}}{\biggr )}\ dx\ .}$

The antiderivative of −1/x² can be found with the power rule and is 1/x.

Alternatively, one may choose u and v such that the product u′ (∫v dx) simplifies due to cancellation. For example, suppose one wishes to integrate:

${\displaystyle \int \sec ^{2}(x)\cdot \ln {\Big (}{\bigl |}\sin(x){\bigr |}{\Big )}\ dx.}$

If we choose u(x) = ln(|sin(x)|) and v(x) = sec²x, then u differentiates to 1/ tan x using the chain rule and v integrates to tan x; so the formula gives:

${\displaystyle \int \sec ^{2}(x)\cdot \ln {\Big (}{\bigl |}\sin(x){\bigr |}{\Big )}\ dx=\tan(x)\cdot \ln {\Big (}{\bigl |}\sin(x){\bigr |}{\Big )}-\int \tan(x)\cdot {\frac {1}{\tan(x)}}\,dx\ .}$

The integrand simplifies to 1, so the antiderivative is x. Finding a simplifying combination frequently involves experimentation.

In some applications, it may not be necessary to ensure that the integral produced by integration by parts has a simple form; for example, in numerical analysis, it may suffice that it has small magnitude and so contributes only a small error term. Some other special techniques are demonstrated in the examples below.

Polynomials and trigonometric functions

In order to calculate

${\displaystyle I=\int x\cos(x)\ dx\ ,}$

let:

${\displaystyle u=x\ \Rightarrow \ du=dx}$

${\displaystyle dv=\cos(x)\ dx\ \Rightarrow \ v=\int \cos(x)\ dx=\sin(x)}$

then:

${\displaystyle {\begin{aligned}\int x\cos(x)\ dx&=\int u\ dv\\&=u\cdot v-\int v\,du\\&=x\sin(x)-\int \sin(x)\ dx\\&=x\sin(x)+\cos(x)+C,\end{aligned}}}$

where C is a constant of integration.

For higher powers of x in the form

${\displaystyle \int x^{n}e^{x}\ dx,\ \int x^{n}\sin(x)\ dx,\ \int x^{n}\cos(x)\ dx\ ,}$

repeatedly using integration by parts can evaluate integrals such as these; each application of the theorem lowers the power of x by one.

Exponentials and trigonometric functions

See also: Integration using Euler's formula

An example commonly used to examine the workings of integration by parts is

${\displaystyle I=\int e^{x}\cos(x)\ dx.}$

Here, integration by parts is performed twice. First let

${\displaystyle u=\cos(x)\ \Rightarrow \ du=-\sin(x)\ dx}$

${\displaystyle dv=e^{x}\ dx\ \Rightarrow \ v=\int e^{x}\ dx=e^{x}}$

then:

${\displaystyle \int e^{x}\cos(x)\ dx=e^{x}\cos(x)+\int e^{x}\sin(x)\ dx.}$

Now, to evaluate the remaining integral, we use integration by parts again, with:

${\displaystyle u=\sin(x)\ \Rightarrow \ du=\cos(x)\ dx}$

${\displaystyle dv=e^{x}\ dx\ \Rightarrow \ v=\int e^{x}\ dx=e^{x}.}$

Then:

${\displaystyle \int e^{x}\sin(x)\ dx=e^{x}\sin(x)-\int e^{x}\cos(x)\ dx.}$

Putting these together,

${\displaystyle \int e^{x}\cos(x)\ dx=e^{x}\cos(x)+e^{x}\sin(x)-\int e^{x}\cos(x)\ dx.}$

The same integral shows up on both sides of this equation. The integral can simply be added to both sides to get

${\displaystyle 2\int e^{x}\cos(x)\ dx=e^{x}{\bigl [}\sin(x)+\cos(x){\bigr ]}+C,}$

which rearranges to

${\displaystyle \int e^{x}\cos(x)\ dx={\frac {1}{2}}e^{x}{\bigl [}\sin(x)+\cos(x){\bigr ]}+C'}$

where again C (and C′ = C/2) is a constant of integration.

A similar method is used to find the integral of secant cubed.

Functions multiplied by unity

Two other well-known examples are when integration by parts is applied to a function expressed as a product of 1 and itself. This works if the derivative of the function is known, and the integral of this derivative times x is also known.

The first example is ∫ ln(x) dx. We write this as:

${\displaystyle I=\int \ln(x)\cdot 1\ dx\ .}$

Let:

${\displaystyle u=\ln(x)\ \Rightarrow \ du={\frac {dx}{x}}}$

${\displaystyle dv=dx\ \Rightarrow \ v=x}$

then:

${\displaystyle {\begin{aligned}\int \ln(x)\ dx&=x\ln(x)-\int {\frac {x}{x}}\ dx\\&=x\ln(x)-\int 1\ dx\\&=x\ln(x)-x+C\end{aligned}}}$

where C is the constant of integration.

The second example is the inverse tangent function arctan(x):

${\displaystyle I=\int \arctan(x)\ dx.}$

Rewrite this as

${\displaystyle \int \arctan(x)\cdot 1\ dx.}$

Now let:

${\displaystyle u=\arctan(x)\ \Rightarrow \ du={\frac {dx}{1+x^{2}}}}$

${\displaystyle dv=dx\ \Rightarrow \ v=x}$

then

${\displaystyle {\begin{aligned}\int \arctan(x)\ dx&=x\arctan(x)-\int {\frac {x}{1+x^{2}}}\ dx\\[8pt]&=x\arctan(x)-{\frac {\ln(1+x^{2})}{2}}+C\end{aligned}}}$

using a combination of the inverse chain rule method and the natural logarithm integral condition.

LIATE rule

A rule of thumb has been proposed, consisting of choosing as u the function that comes first in the following list:^[4]

L – logarithmic functions: ${\displaystyle \ln(x),\ \log _{b}(x),}$ etc.

I – inverse trigonometric functions (including hyperbolic analogues): ${\displaystyle \arctan(x),\ \operatorname {arcsec}(x),\ \operatorname {arsinh} (x),}$ etc.

A – algebraic functions: ${\displaystyle x^{2},\ 3x^{50},}$ etc.

T – trigonometric functions (including hyperbolic analogues): ${\displaystyle \sin(x),\ \tan(x),\ \operatorname {sech} (x),}$ etc.

E – exponential functions: ${\displaystyle e^{x},\ 19^{x},}$ etc.

The function which is to be dv is whichever comes last in the list. The reason is that functions lower on the list generally have easier antiderivatives than the functions above them. The rule is sometimes written as "DETAIL" where D stands for dv and the top of the list is the function chosen to be dv.

To demonstrate the LIATE rule, consider the integral

${\displaystyle \int x\cdot \cos(x)\,dx.}$

Following the LIATE rule, u = x, and dv = cos(x) dx, hence du = dx, and v = sin(x), which makes the integral become

${\displaystyle x\cdot \sin(x)-\int 1\sin(x)\,dx,}$

which equals

${\displaystyle x\cdot \sin(x)+\cos(x)+C.}$

In general, one tries to choose u and dv such that du is simpler than u and dv is easy to integrate. If instead cos(x) was chosen as u, and x dx as dv, we would have the integral

${\displaystyle {\frac {x^{2}}{2}}\cos(x)+\int {\frac {x^{2}}{2}}\sin(x)\,dx,}$

which, after recursive application of the integration by parts formula, would clearly result in an infinite recursion and lead nowhere.

Although a useful rule of thumb, there are exceptions to the LIATE rule. A common alternative is to consider the rules in the "ILATE" order instead. Also, in some cases, polynomial terms need to be split in non-trivial ways. For example, to integrate

${\displaystyle \int x^{3}e^{x^{2}}\,dx,}$

one would set

${\displaystyle u=x^{2},\quad dv=x\cdot e^{x^{2}}\,dx,}$

so that

${\displaystyle du=2x\,dx,\quad v={\frac {e^{x^{2}}}{2}}.}$

Then

${\displaystyle \int x^{3}e^{x^{2}}\,dx=\int \left(x^{2}\right)\left(xe^{x^{2}}\right)\,dx=\int u\,dv=uv-\int v\,du={\frac {x^{2}e^{x^{2}}}{2}}-\int xe^{x^{2}}\,dx.}$

Finally, this results in

${\displaystyle \int x^{3}e^{x^{2}}\,dx={\frac {e^{x^{2}}\left(x^{2}-1\right)}{2}}+C.}$

Integration by parts is often used as a tool to prove theorems in mathematical analysis.

Wallis product

The Wallis infinite product for ${\displaystyle \pi }$

${\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)\\[6pt]&={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdot \;\cdots \end{aligned}}}$

may be derived using integration by parts.

Gamma function identity

The gamma function is an example of a special function, defined as an improper integral for ${\displaystyle z>0}$. Integration by parts illustrates it to be an extension of the factorial function:

${\displaystyle {\begin{aligned}\Gamma (z)&=\int _{0}^{\infty }e^{-x}x^{z-1}dx\\[6pt]&=-\int _{0}^{\infty }x^{z-1}\,d\left(e^{-x}\right)\\[6pt]&=-{\Biggl [}e^{-x}x^{z-1}{\Biggl ]}_{0}^{\infty }+\int _{0}^{\infty }e^{-x}d\left(x^{z-1}\right)\\[6pt]&=0+\int _{0}^{\infty }\left(z-1\right)x^{z-2}e^{-x}dx\\[6pt]&=(z-1)\Gamma (z-1).\end{aligned}}}$

Since

${\displaystyle \Gamma (1)=\int _{0}^{\infty }e^{-x}\,dx=1,}$

when ${\displaystyle z}$ is a natural number, that is, ${\displaystyle z=n\in \mathbb {N} }$, applying this formula repeatedly gives the factorial: ${\displaystyle \Gamma (n+1)=n!}$

Use in harmonic analysis

Integration by parts is often used in harmonic analysis, particularly Fourier analysis, to show that quickly oscillating integrals with sufficiently smooth integrands decay quickly. The most common example of this is its use in showing that the decay of function's Fourier transform depends on the smoothness of that function, as described below.

Fourier transform of derivative

If f is a k-times continuously differentiable function and all derivatives up to the kth one decay to zero at infinity, then its Fourier transform satisfies

${\displaystyle ({\mathcal {F}}f^{(k)})(\xi )=(2\pi i\xi )^{k}{\mathcal {F}}f(\xi ),}$

where f^(k) is the kth derivative of f. (The exact constant on the right depends on the convention of the Fourier transform used.) This is proved by noting that

${\displaystyle {\frac {d}{dy}}e^{-2\pi iy\xi }=-2\pi i\xi e^{-2\pi iy\xi },}$

so using integration by parts on the Fourier transform of the derivative we get

${\displaystyle {\begin{aligned}({\mathcal {F}}f')(\xi )&=\int _{-\infty }^{\infty }e^{-2\pi iy\xi }f'(y)\,dy\\&=\left[e^{-2\pi iy\xi }f(y)\right]_{-\infty }^{\infty }-\int _{-\infty }^{\infty }(-2\pi i\xi e^{-2\pi iy\xi })f(y)\,dy\\[5pt]&=2\pi i\xi \int _{-\infty }^{\infty }e^{-2\pi iy\xi }f(y)\,dy\\[5pt]&=2\pi i\xi {\mathcal {F}}f(\xi ).\end{aligned}}}$

Applying this inductively gives the result for general k. A similar method can be used to find the Laplace transform of a derivative of a function.

Decay of Fourier transform

The above result tells us about the decay of the Fourier transform, since it follows that if f and f^(k) are integrable then

${\displaystyle \vert {\mathcal {F}}f(\xi )\vert \leq {\frac {I(f)}{1+\vert 2\pi \xi \vert ^{k}}},{\text{ where }}I(f)=\int _{-\infty }^{\infty }{\Bigl (}\vert f(y)\vert +\vert f^{(k)}(y)\vert {\Bigr )}\,dy.}$

In other words, if f satisfies these conditions then its Fourier transform decays at infinity at least as quickly as 1/|ξ|^k. In particular, if k ≥ 2 then the Fourier transform is integrable.

The proof uses the fact, which is immediate from the definition of the Fourier transform, that

${\displaystyle \vert {\mathcal {F}}f(\xi )\vert \leq \int _{-\infty }^{\infty }\vert f(y)\vert \,dy.}$

Using the same idea on the equality stated at the start of this subsection gives

${\displaystyle \vert (2\pi i\xi )^{k}{\mathcal {F}}f(\xi )\vert \leq \int _{-\infty }^{\infty }\vert f^{(k)}(y)\vert \,dy.}$

Summing these two inequalities and then dividing by 1 + |2πξ^k| gives the stated inequality.

Use in operator theory

One use of integration by parts in operator theory is that it shows that the −∆ (where ∆ is the Laplace operator) is a positive operator on L² (see L^p space). If f is smooth and compactly supported then, using integration by parts, we have

${\displaystyle {\begin{aligned}\langle -\Delta f,f\rangle _{L^{2}}&=-\int _{-\infty }^{\infty }f''(x){\overline {f(x)}}\,dx\\[5pt]&=-\left[f'(x){\overline {f(x)}}\right]_{-\infty }^{\infty }+\int _{-\infty }^{\infty }f'(x){\overline {f'(x)}}\,dx\\[5pt]&=\int _{-\infty }^{\infty }\vert f'(x)\vert ^{2}\,dx\geq 0.\end{aligned}}}$

Other applications

• Determining boundary conditions in Sturm–Liouville theory
• Deriving the Euler–Lagrange equation in the calculus of variations

Repeated integration by parts

Considering a second derivative of ${\displaystyle v}$ in the integral on the LHS of the formula for partial integration suggests a repeated application to the integral on the RHS:

${\displaystyle \int uv''\,dx=uv'-\int u'v'\,dx=uv'-\left(u'v-\int u''v\,dx\right).}$

Extending this concept of repeated partial integration to derivatives of degree n leads to

${\displaystyle {\begin{aligned}\int u^{(0)}v^{(n)}\,dx&=u^{(0)}v^{(n-1)}-u^{(1)}v^{(n-2)}+u^{(2)}v^{(n-3)}-\cdots +(-1)^{n-1}u^{(n-1)}v^{(0)}+(-1)^{n}\int u^{(n)}v^{(0)}\,dx.\\[5pt]&=\sum _{k=0}^{n-1}(-1)^{k}u^{(k)}v^{(n-1-k)}+(-1)^{n}\int u^{(n)}v^{(0)}\,dx.\end{aligned}}}$

This concept may be useful when the successive integrals of ${\displaystyle v^{(n)}}$ are readily available (e.g., plain exponentials or sine and cosine, as in Laplace or Fourier transforms), and when the nth derivative of ${\displaystyle u}$ vanishes (e.g., as a polynomial function with degree ${\displaystyle (n-1)}$). The latter condition stops the repeating of partial integration, because the RHS-integral vanishes.

In the course of the above repetition of partial integrations the integrals

${\displaystyle \int u^{(0)}v^{(n)}\,dx\quad }$ and ${\displaystyle \quad \int u^{(\ell )}v^{(n-\ell )}\,dx\quad }$ and ${\displaystyle \quad \int u^{(m)}v^{(n-m)}\,dx\quad {\text{ for }}1\leq m,\ell \leq n}$

get related. This may be interpreted as arbitrarily "shifting" derivatives between ${\displaystyle v}$ and ${\displaystyle u}$ within the integrand, and proves useful, too (see Rodrigues' formula).

Tabular integration by parts

The essential process of the above formula can be summarized in a table; the resulting method is called "tabular integration" and was featured in the film Stand and Deliver.

For example, consider the integral

${\displaystyle \int x^{3}\cos x\,dx\quad }$ and take ${\displaystyle \quad u^{(0)}=x^{3},\quad v^{(n)}=\cos x.}$

Begin to list in column A the function ${\displaystyle u^{(0)}=x^{3}}$ and its subsequent derivatives ${\displaystyle u^{(i)}}$ until zero is reached. Then list in column B the function ${\displaystyle v^{(n)}=\cos x}$ and its subsequent integrals ${\displaystyle v^{(n-i)}}$ until the size of column B is the same as that of column A. The result is as follows:

# i	Sign	A: derivatives u(i)	B: integrals v(n−i)
0	+	${\displaystyle x^{3}}$	${\displaystyle \cos x}$
1	−	${\displaystyle 3x^{2}}$	${\displaystyle \sin x}$
2	+	${\displaystyle 6x}$	${\displaystyle -\cos x}$
3	−	${\displaystyle 6}$	${\displaystyle -\sin x}$
4	+	${\displaystyle 0}$	${\displaystyle \cos x}$

The product of the entries in row i of columns A and B together with the respective sign give the relevant integrals in step i in the course of repeated integration by parts. Step i = 0 yields the original integral. For the complete result in step i > 0 the ith integral must be added to all the previous products (0 ≤ j < i) of the jth entry of column A and the (j + 1)st entry of column B (i.e., multiply the 1st entry of column A with the 2nd entry of column B, the 2nd entry of column A with the 3rd entry of column B, etc. ...) with the given jth sign. This process comes to a natural halt, when the product, which yields the integral, is zero (i = 4 in the example). The complete result is the following (with the alternating signs in each term):

${\displaystyle \underbrace {(+1)(x^{3})(\sin x)} _{j=0}+\underbrace {(-1)(3x^{2})(-\cos x)} _{j=1}+\underbrace {(+1)(6x)(-\sin x)} _{j=2}+\underbrace {(-1)(6)(\cos x)} _{j=3}+\underbrace {\int (+1)(0)(\cos x)\,dx} _{i=4:\;\to \;C}.}$

This yields

${\displaystyle \underbrace {\int x^{3}\cos x\,dx} _{\text{step 0}}=x^{3}\sin x+3x^{2}\cos x-6x\sin x-6\cos x+C.}$

The repeated partial integration also turns out useful, when in the course of respectively differentiating and integrating the functions ${\displaystyle u^{(i)}}$ and ${\displaystyle v^{(n-i)}}$ their product results in a multiple of the original integrand. In this case the repetition may also be terminated with this index i.This can happen, expectably, with exponentials and trigonometric functions. As an example consider

${\displaystyle \int e^{x}\cos x\,dx.}$

# i	Sign	A: derivatives u(i)	B: integrals v(n−i)
0	+	${\displaystyle e^{x}}$	${\displaystyle \cos x}$
1	−	${\displaystyle e^{x}}$	${\displaystyle \sin x}$
2	+	${\displaystyle e^{x}}$	${\displaystyle -\cos x}$

In this case the product of the terms in columns A and B with the appropriate sign for index i = 2 yields the negative of the original integrand (compare rows i = 0 and i = 2).

${\displaystyle \underbrace {\int e^{x}\cos x\,dx} _{\text{step 0}}=\underbrace {(+1)(e^{x})(\sin x)} _{j=0}+\underbrace {(-1)(e^{x})(-\cos x)} _{j=1}+\underbrace {\int (+1)(e^{x})(-\cos x)\,dx} _{i=2}.}$

Observing that the integral on the RHS can have its own constant of integration ${\displaystyle C'}$, and bringing the abstract integral to the other side, gives

${\displaystyle 2\int e^{x}\cos x\,dx=e^{x}\sin x+e^{x}\cos x+C',}$

and finally:

${\displaystyle \int e^{x}\cos x\,dx={\frac {1}{2}}\left(e^{x}(\sin x+\cos x)\right)+C,}$

where C = C′/2.

Higher dimensions

Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V.

The product rule for divergence states:

$${\displaystyle \nabla \cdot (u\mathbf {V} )\ =\ u\,\nabla \cdot \mathbf {V} \ +\ \nabla u\cdot \mathbf {V} .}$$

Suppose ${\displaystyle \Omega }$ is an open bounded subset of ${\displaystyle \mathbb {R} ^{n}}$ with a piecewise smooth boundary ${\displaystyle \Gamma =\partial \Omega }$. Integrating over ${\displaystyle \Omega }$ with respect to the standard volume form ${\displaystyle d\Omega }$, and applying the divergence theorem, gives:

$${\displaystyle \int _{\Gamma }u\mathbf {V} \cdot {\hat {\mathbf {n} }}\,d\Gamma \ =\ \int _{\Omega }\nabla \cdot (u\mathbf {V} )\,d\Omega \ =\ \int _{\Omega }u\,\nabla \cdot \mathbf {V} \,d\Omega \ +\ \int _{\Omega }\nabla u\cdot \mathbf {V} \,d\Omega ,}$$

where ${\displaystyle {\hat {\mathbf {n} }}}$ is the outward unit normal vector to the boundary, integrated with respect to its standard Riemannian volume form ${\displaystyle d\Gamma }$. Rearranging gives:

$${\displaystyle \int _{\Omega }u\,\nabla \cdot \mathbf {V} \,d\Omega \ =\ \int _{\Gamma }u\mathbf {V} \cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }\nabla u\cdot \mathbf {V} \,d\Omega ,}$$

or in other words

$${\displaystyle \int _{\Omega }u\,\operatorname {div} (\mathbf {V} )\,d\Omega \ =\ \int _{\Gamma }u\mathbf {V} \cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }\operatorname {grad} (u)\cdot \mathbf {V} \,d\Omega .}$$

The regularity requirements of the theorem can be relaxed. For instance, the boundary ${\displaystyle \Gamma =\partial \Omega }$ need only be Lipschitz continuous, and the functions u, v need only lie in the Sobolev space H¹(Ω).

Green's first identity

Consider the continuously differentiable vector fields ${\displaystyle \mathbf {U} =u_{1}\mathbf {e} _{1}+\cdots +u_{n}\mathbf {e} _{n}}$ and ${\displaystyle v\mathbf {e} _{1},\ldots ,v\mathbf {e} _{n}}$, where ${\displaystyle \mathbf {e} _{i}}$is the i-th standard basis vector for ${\displaystyle i=1,\ldots ,n}$. Now apply the above integration by parts to each ${\displaystyle u_{i}}$ times the vector field ${\displaystyle v\mathbf {e} _{i}}$:

$${\displaystyle \int _{\Omega }u_{i}{\frac {\partial v}{\partial x_{i}}}\,d\Omega \ =\ \int _{\Gamma }u_{i}v\,\mathbf {e} _{i}\cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }{\frac {\partial u_{i}}{\partial x_{i}}}v\,d\Omega .}$$

Summing over i gives a new integration by parts formula:

$${\displaystyle \int _{\Omega }\mathbf {U} \cdot \nabla v\,d\Omega \ =\ \int _{\Gamma }v\mathbf {U} \cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }v\,\nabla \cdot \mathbf {U} \,d\Omega .}$$

The case ${\displaystyle \mathbf {U} =\nabla u}$, where ${\displaystyle u\in C^{2}({\bar {\Omega }})}$, is known as the first of Green's identities:

$${\displaystyle \int _{\Omega }\nabla u\cdot \nabla v\,d\Omega \ =\ \int _{\Gamma }v\,\nabla u\cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }v\,\nabla ^{2}u\,d\Omega .}$$