Integration by parts

Incalculus,and more generally inmathematical analysis,integration by partsorpartial integrationis a process that finds theintegralof aproductoffunctionsin terms of the integral of the product of theirderivativeandantiderivative.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 theproduct ruleofdifferentiation;it is indeed derived using the product rule.

The integration by parts formula states: $${\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}$$

Or, letting$u=u(x)$and$du=u'(x)\,dx$while$v=v(x)$and$dv=v'(x)\,dx,$the formula can be written more compactly: $$\int u\,dv\ =\ uv-\int v\,du.$$

It is important to note that the former expression is written as a definite integral and the latter is written as an indefinite integral. Applying the appropriate limits to the latter expression should yield the former, but the latter is not necessarily equivalent to the former.

MathematicianBrook Taylordiscovered integration by parts, first publishing the idea in 1715.More general formulations of integration by parts exist for theRiemann–StieltjesandLebesgue–Stieltjes integrals.Thediscreteanalogue forsequencesis calledsummation by parts.

Theorem

Product of two functions

The theorem can be derived as follows. For twocontinuously differentiablefunctions$u(x)$and$v(x)$,theproduct rulestates:

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

Integrating both sides with respect to$x$,

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

and noting that anindefinite integralis an antiderivative gives

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

where we neglect writing theconstant of integration.This yields the formula forintegration by parts:

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

or in terms of thedifferentials$du=u'(x)\,dx$,$dv=v'(x)\,dx,\quad$

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

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 thefundamental theorem of calculusgives the definite integral version: $$\int _{a}^{b}u(x)v'(x)\,dx=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.$$ The original integral$\int uv'\,dx$contains thederivativev';to apply the theorem, one must findv,theantiderivativeofv',then evaluate the resulting integral$\int vu'\,dx.$

Validity for less smooth functions

It is not necessary for$u$and$v$to be continuously differentiable. Integration by parts works if$u$isabsolutely continuousand the function designated$v'$isLebesgue 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 notcompact,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

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

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

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

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

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

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

$$\int _{1}^{\infty }u(x)v'(x)\,dx={\Big [}u(x)v(x){\Big ]}_{1}^{\infty }-\int _{1}^{\infty }u'(x)v(x)\,dx$$ with the same interpretation.

One can also easily come up with similar examples in which$u$and$v$arenotcontinuously differentiable.

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

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

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

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

In general, for$n$factors

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

which leads to

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

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 locallyone-to-oneandintegrable,we can define $${\begin{aligned}x(y)&=f(g^{-1}(y))\\y(x)&=g(f^{-1}(x))\end{aligned}}$$

The area of the blue region is

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

Similarly, the area of the red region is $$A_{2}=\int _{x_{1}}^{x_{2}}y(x)\,dx$$

The total areaA₁+A₂is equal to the area of the bigger rectangle,x₂y₂,minus the area of the smaller one,x₁y₁:

$$\overbrace {\int _{y_{1}}^{y_{2}}x(y)\,dy} ^{A_{1}}+\overbrace {\int _{x_{1}}^{x_{2}}y(x)\,dx} ^{A_{2}}\ =\ {\biggl.}x\cdot y(x){\biggl |}_{x_{1}}^{x_{2}}\ =\ {\biggl.}y\cdot x(y){\biggl |}_{y_{1}}^{y_{2}}$$ Or, in terms oft, $$\int _{t_{1}}^{t_{2}}x(t)\,dy(t)+\int _{t_{1}}^{t_{2}}y(t)\,dx(t)\ =\ {\biggl.}x(t)y(t){\biggl |}_{t_{1}}^{t_{2}}$$ Or, in terms of indefinite integrals, this can be written as $$\int x\,dy+\int y\,dx\ =\ xy$$ Rearranging: $$\int x\,dy\ =\ xy-\int y\,dx$$ 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 functionf⁻¹(x) when the integral of the functionf(x) is known. Indeed, the functionsx(y) andy(x) are inverses, and the integral ∫xdymay be calculated as above from knowing the integral ∫ydx.In particular, this explains use of integration by parts to integratelogarithmandinverse trigonometric functions.In fact, if$f$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$f^{-1}$in terms of the integral of$f$.This is demonstrated in the article,Integral of inverse functions.

Applications

Finding antiderivatives

Integration by parts is aheuristicrather 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 functionsu(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:

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

On the right-hand side,uis differentiated andvis integrated; consequently it is useful to chooseuas a function that simplifies when differentiated, or to choosevas a function that simplifies when integrated. As a simple example, consider:

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

Since the derivative of ln(x) is1/x,one makes (ln(x)) partu;since the antiderivative of1/x²is −1/x,one makes1/x²partv.The formula now yields:

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

The antiderivative of −1/x²can be found with thepower ruleand is1/x.

Alternatively, one may chooseuandvsuch that the productu′ (∫vdx) simplifies due to cancellation. For example, suppose one wishes to integrate:

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

If we chooseu(x) = ln(|sin(x)|) andv(x) = sec²x, thenudifferentiates to 1/ tanxusing thechain ruleandvintegrates to tanx;so the formula gives:

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

The integrand simplifies to 1, so the antiderivative isx.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, innumerical 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

$$I=\int x\cos(x)\,dx\,,$$

let: $${\begin{alignedat}{3}u&=x\ &\Rightarrow \ &&du&=dx\\dv&=\cos(x)\,dx\ &\Rightarrow \ &&v&=\int \cos(x)\,dx=\sin(x)\end{alignedat}}$$

then:

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

whereCis aconstant of integration.

For higher powers of$x$in the form

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

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

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

Here, integration by parts is performed twice. First let

$${\begin{alignedat}{3}u&=\cos(x)\ &\Rightarrow \ &&du&=-\sin(x)\,dx\\dv&=e^{x}\,dx\ &\Rightarrow \ &&v&=\int e^{x}\,dx=e^{x}\end{alignedat}}$$

then:

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

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

$${\begin{alignedat}{3}u&=\sin(x)\ &\Rightarrow \ &&du&=\cos(x)\,dx\\dv&=e^{x}\,dx\,&\Rightarrow \ &&v&=\int e^{x}\,dx=e^{x}.\end{alignedat}}$$

Then:

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

Putting these together,

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

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

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

which rearranges to

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

where again$C$(and$C'=C/2$) is aconstant of integration.

A similar method is used to find theintegral 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$\int \ln(x)dx$.We write this as:

$$I=\int \ln(x)\cdot 1\,dx\,.$$

Let:

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

then:

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

where$C$is theconstant of integration.

The second example is theinverse tangentfunction$\arctan(x)$:

$$I=\int \arctan(x)\,dx.$$

Rewrite this as

$$\int \arctan(x)\cdot 1\,dx.$$

Now let:

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

$$dv=dx\ \Rightarrow \ v=x$$

then

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

using a combination of theinverse chain rule methodand thenatural logarithm integral condition.

LIATE rule

The LIATE rule is a rule of thumb for integration by parts. It involves choosing asuthe function that comes first in the following list:

• L–logarithmic functions:$\ln(x),\ \log _{b}(x),$etc.
• I–inverse trigonometric functions(includinghyperbolic analogues):$\arctan(x),\ \operatorname {arcsec}(x),\ \operatorname {arsinh} (x),$etc.
• A–algebraic functions(such aspolynomials):$x^{2},\ 3x^{50},$etc.
• T–trigonometric functions(includinghyperbolic analogues):$\sin(x),\ \tan(x),\ \operatorname {sech} (x),$etc.
• E–exponential functions:$e^{x},\ 19^{x},$etc.

The function which is to bedvis whichever comes last in the list. The reason is that functions lower on the list generally have simplerantiderivativesthan the functions above them. The rule is sometimes written as "DETAIL", whereDstands fordvand the top of the list is the function chosen to bedv.An alternative to this rule is the ILATE rule, where inverse trigonometric functions come before logarithmic functions.

To demonstrate the LIATE rule, consider the integral

$$\int x\cdot \cos(x)\,dx.$$

Following the LIATE rule,u=x,anddv= cos(x) dx,hencedu=dx,andv= sin(x), which makes the integral become $$x\cdot \sin(x)-\int 1\sin(x)\,dx,$$ which equals $$x\cdot \sin(x)+\cos(x)+C.$$

In general, one tries to chooseuanddvsuch thatduis simpler thanuanddvis easy to integrate. If instead cos(x) was chosen asu,andx dxasdv,we would have the integral

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

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

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

one would set

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

so that

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

Then

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

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

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

Wallis product

The Wallis infinite product for$\pi$

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

may bederived using integration by parts.

Gamma function identity

Thegamma functionis an example of aspecial function,defined as animproper integralfor$z>0$.Integration by parts illustrates it to be an extension of the factorial function:

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

Since

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

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

Use in harmonic analysis

Integration by parts is often used inharmonic analysis,particularlyFourier analysis,to showthat 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$k$th one decay to zero at infinity, then itsFourier transformsatisfies

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

where$f^{(k)}$is the$k$th derivative of$f$.(The exact constant on the right depends on theconvention of the Fourier transform used.) This is proved by noting that

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

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

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

Applying thisinductivelygives the result for general$k$.A similar method can be used to find theLaplace transformof 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

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

In other words, if$f$satisfies these conditions then its Fourier transform decays at infinity at least as quickly as1/|ξ|^k.In particular, if$k\geq 2$then the Fourier transform is integrable.

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

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

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

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

Summing these two inequalities and then dividing by1 + |2πξ^k|gives the stated inequality.

Use in operator theory

One use of integration by parts inoperator theoryis that it shows that the−∆(where ∆ is theLaplace operator) is apositive operatoron$L^{2}$(seeL^pspace). If$f$is smooth and compactly supported then, using integration by parts, we have

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

Other applications

• Determiningboundary conditionsinSturm–Liouville theory
• Deriving theEuler–Lagrange equationin thecalculus of variations

Repeated integration by parts

Considering a second derivative of$v$in the integral on the LHS of the formula for partial integration suggests a repeated application to the integral on the RHS: $$\int uv''\,dx=uv'-\int u'v'\,dx=uv'-\left(u'v-\int u''v\,dx\right).$$

Extending this concept of repeated partial integration to derivatives of degreenleads to $${\begin{aligned}\int u^{(0)}v^{(n)}\,dx&=u^{(0)}v^{(n-1)}-u^{(1)}v^{(n-2)}+u^{(2)}v^{(n-3)}-\cdots +(-1)^{n-1}u^{(n-1)}v^{(0)}+(-1)^{n}\int u^{(n)}v^{(0)}\,dx.\\[5pt]&=\sum _{k=0}^{n-1}(-1)^{k}u^{(k)}v^{(n-1-k)}+(-1)^{n}\int u^{(n)}v^{(0)}\,dx.\end{aligned}}$$

This concept may be useful when the successive integrals of$v^{(n)}$are readily available (e.g., plain exponentials or sine and cosine, as inLaplaceorFourier transforms), and when thenth derivative of$u$vanishes (e.g., as a polynomial function with degree$(n-1)$). 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 $$\int u^{(0)}v^{(n)}\,dx\quad$$and$\quad \int u^{(\ell )}v^{(n-\ell )}\,dx\quad$and$\quad \int u^{(m)}v^{(n-m)}\,dx\quad {\text{ for }}1\leq m,\ell \leq n$ get related. This may be interpreted as arbitrarily "shifting" derivatives between$v$and$u$within the integrand, and proves useful, too (seeRodrigues' 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 filmStand and Deliver(1988).

For example, consider the integral

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

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

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

The product of the entries inrowiof columnsAandBtogether with the respective sign give the relevant integrals instepiin the course of repeated integration by parts.Stepi= 0yields the original integral. For the complete result instepi> 0theith integralmust be added to all the previous products (0 ≤j<i) of thejth entryof column A and the(j+ 1)st entryof 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 givenjth sign.This process comes to a natural halt, when the product, which yields the integral, is zero (i= 4in the example). The complete result is the following (with the alternating signs in each term):

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

This yields

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

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

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

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

In this case the product of the terms in columnsAandBwith the appropriate sign for indexi= 2yields the negative of the original integrand (comparerowsi= 0andi= 2).

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

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

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

and finally:

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

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 functionuand vector-valued function (vector field)V.

Theproduct rule for divergencestates:

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

Suppose$\Omega$is anopenbounded subsetof$\mathbb {R} ^{n}$with apiecewise smoothboundary$\Gamma =\partial \Omega$.Integrating over$\Omega$with respect to the standard volume form$d\Omega$,and applying thedivergence theorem,gives:

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

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

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

or in other words $$\int _{\Omega }u\,\operatorname {div} (\mathbf {V} )\,d\Omega \ =\ \int _{\Gamma }u\mathbf {V} \cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }\operatorname {grad} (u)\cdot \mathbf {V} \,d\Omega.$$ Theregularityrequirements of the theorem can be relaxed. For instance, the boundary$\Gamma =\partial \Omega$need only beLipschitz continuous,and the functionsu,vneed only lie in theSobolev space$H^{1}(\Omega )$.

Green's first identity

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

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

Summing overigives a new integration by parts formula:

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

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

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

See also

• Integration by parts for the Lebesgue–Stieltjes integral
• Integration by partsforsemimartingales,involving their quadratic covariation.
• Integration by substitution
• Legendre transformation

Further reading

• Louis Brand (10 October 2013).Advanced Calculus: An Introduction to Classical Analysis.Courier Corporation. pp. 267–.ISBN978-0-486-15799-3.
• Hoffmann, Laurence D.; Bradley, Gerald L. (2004).Calculus for Business, Economics, and the Social and Life Sciences(8th ed.). pp. 450–464.ISBN0-07-242432-X.
• Willard, Stephen (1976).Calculus and its Applications.Boston: Prindle, Weber & Schmidt. pp. 193–214.ISBN0-87150-203-8.
• Washington, Allyn J. (1966).Technical Calculus with Analytic Geometry.Reading: Addison-Wesley. pp. 218–245.ISBN0-8465-8603-7.

External links

• "Integration by parts",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Integration by parts—from MathWorld