Integration by parts

Topics in Calculus

Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities: Power rule, Product rule, Quotient rule, Chain rule Integral calculus  Integral Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order Vector calculus  Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem Multivariable calculus  Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other (ideally simpler) integrals. The rule can be derived in one line by simply integrating the product rule of differentiation.

If u = f(x), v = g(x), and the differentials du = f '(x) dx and dv = g'(x) dx, then integration by parts states that

$\int f(x) g'(x)\, dx = f(x) g(x) - \int f'(x) g(x)\, dx\!$

or simply:

$\int u\, dv=uv-\int v\, du\!$

Rule

Suppose f(x) and g(x) are two continuously differentiable functions. The product rule states

$(f(x)g(x))' = f(x) g'(x) + f'(x) g(x)\!$

Integrating both sides gives

$f(x) g(x) = \int f(x) g'(x)\, dx + \int f'(x) g(x)\, dx\!$

Rearranging terms

$\int f(x) g'(x)\, dx = f(x) g(x) - \int f'(x) g(x)\, dx\!$

From the above one can derive the integration by parts rule, which states that, given an interval with endpoints a and b,

$\int_a^b f(x) g'(x)\, dx = \left[ f(x) g(x) \right]_a^b - \int_a^b f'(x) g(x)\, dx\!$

with the common notation

$\left[ f(x) g(x) \right]_a^b = f(b) g(b) - f(a) g(a).\!$

The rule is shown to be true by using the product rule for derivatives and the fundamental theorem of calculus. Thus

$\begin{align} f(b)g(b) - f(a)g(a) & = \int_a^b \frac{d}{dx} ( f(x) g(x) )\, dx \\ & = \int_a^b f'(x) g(x) \, dx + \int_a^b f(x) g'(x)\, dx. \end{align}$

In the traditional calculus curriculum, the rule is often stated using indefinite integrals in the form

$\int f(x) g'(x)\, dx = f(x) g(x) - \int f'(x) g(x)\, dx\!$

or, if u = f(x), v = g(x) and the differentials du = f ′(x) dx and dv = g′(x) dx, then it is in the form most often seen:

$\int u\, dv=uv-\int v\, du.\!$

This formula can be interpreted to mean that the area under the graph of a function u(v) is the same as the area of the rectangle u v minus the area above the graph.

The original integral contains the derivative of g; in order to be able to apply the rule, the antiderivative g must be found, and then the resulting integral ∫g f ′ dx must be evaluated.

One can also formulate a discrete analogue for sequences, called summation by parts.

An alternative notation has the advantage that the factors of the original expression are identified as f(x) and g(x):

$\int_a^b f(x) g(x)\, dx = \left[ f(x) \int g(x)\, dx \right]_a^b - \int_a^b \left ( \int g(x)\, dx \right )\, f'(x) \, dx.\!$

This formula is valid whenever f(x) is continuously differentiable and g(x) is continuous.

More general formulations of integration by parts exist for the Riemann–Stieltjes integral and Lebesgue–Stieltjes integral.

More complicated forms of the rule are also valid:

$\int u v\, dw = u v w - \int u w\, dv - \int v w\, du.\!$

Strategy

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 it into a product of two functions ƒ(x)g(x) such that the integral produced by the integration by parts formula is easier to evaluate than the original one. The following form is useful in illustrating the best strategy to take:

$\int f g\, dx = f \int g\, dx - \int \left ( f' \int g\,dx \right )\, dx.\!$

Note that on the right-hand side, ƒ is differentiated and g is integrated; consequently it is useful to choose ƒ as a function that simplifies when differentiated, and/or to choose g as a function that simplifies when integrated. As a simple example, consider:

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

Since the derivative of ln x is 1/x, we make this part of ƒ; since the anti-derivative of 1/x² is −1/x, we make this part of g. The formula now yields:

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

The remaining integral of −1/x² can be completed with the power rule and is 1/x.

Alternatively, we may choose ƒ and g such that the product $f'(\int g\, dx)\!$ simplifies due to cancellation. For example, suppose we wish to integrate:

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

If we choose ƒ(x) = ln(sin x) and g(x) = 1/(cos x)², then ƒ differentiates to 1/tan x using the chain rule and g integrates to tan x; so the formula gives:

$\int \frac{\ln(\sin x)}{(\cos x)^2}\, dx = \ln(\sin x)\tan x - \int \frac{1}{\tan x} \tan x\, dx.\!$

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.

Examples

Integrals with powers of x or e^x

In order to calculate:

$\int x\cos (x) \, dx\!$

Let:

$u = x \Rightarrow du = dx$

$dv = \cos(x)\,dx \Rightarrow v = \int\cos(x)\,dx = \sin x$

Then:

$\begin{align} \int x\cos (x) \,dx & = \int u \, dv \\ & = uv - \int v \, du \\ & = x\sin (x) - \int \sin (x) \, dx \\ & = x\sin (x) + \cos (x) + C. \end{align} \!$

where C is an arbitrary constant of integration.

By repeatedly using integration by parts, integrals such as

$\int x^{3} \sin (x) \, dx \quad \mbox{and} \quad \int x^{2} e^{x} \, dx$

can be computed in the same fashion: each application of the rule lowers the power of x by one.

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

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

Here, integration by parts is performed twice. First let

$u = \cos(x) \Rightarrow du = -\sin(x)\,dx$

and

$dv = e^x \, dx \Rightarrow v = \int e^x \, dx = e^x$

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:

$u = \sin(x) \Rightarrow du = \cos(x)\, dx$

$dv = e^x \, dx \Rightarrow v = \int e^x \, dx = e^x$

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

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

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

where, again, C (and C' = C/2) is an arbitrary constant of integration.

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

Interchange of the order of integration

Main article: Order of integration (calculus)

Integration over the triangular area can be done using vertical or horizontal strips as the first step. This is an overhead view, looking down the z-axis onto the xy-plane. The sloped line is the curve y = x.

For an example of the following type of notation see Multiple_integral#Formulae_of_reduction.

The above formulation includes the technique of interchange of the order of integration, which is not usually viewed in this manner. Consider the iterated integral:

$\int_a^z \ dx\ \int_a^x \ dy \ h(y).$

In the order written above, the strip of width dx is integrated first over the y-direction (a strip of width dx in the x direction is integrated with respect to the y variable across the y direction) as shown in the left panel of the figure, which is inconvenient especially when function h(y) is not easily integrated. The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z, and then the result is integrated from y = a to y = z, resulting in:

$\int_a^z \ dx\ \int_a^x \ dy \ h(y) = \int_a^z \ dy \ h(y)\ \int_y^z \ dx = \int_a^z \ dy \ \left(z-y\right) h(y) \ .$

This result can be seen to be an example of the above formula for integration by parts, repeated below:

$\int_a^z f(x) g'(x)\, dx = \left[ f(x) g(x) \right]_{a}^{z} - \int_a^z f'(x) g(x)\, dx\!$

Substitute:

$f (x) = \int_a^x \ h(y)\, dy \text{ and }g'(x) = 1 \ .$

Which gives the result.

More examples

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:

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

Let:

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

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

Then:

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

where, again, C is the constant of integration.

The second example is ∫ arctan(x) dx, where arctan(x) is the inverse tangent function. 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

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

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

Here is an example:

$\int x\, dx = x^2 - \int x\, dx\!$

$2 \int x\, dx = x^2\!$

$\int x\, dx = \frac{x^2}{2} + C\!$

Liate rule

A rule of thumb proposed by Herbert Kasube of Bradley University advises that whichever function comes first in the following list should be u:^[1]

L: Logarithmic functions: ln x, log_b x, etc.

I: Inverse trigonometric functions: arctan x, arcsec x, etc.

A: Algebraic functions: x², 3x⁵⁰, etc.

T: Trigonometric functions: sin x, tan x, etc.

E: Exponential functions: e^x, 19^x, etc.

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

To demonstrate the LIATE rule, consider the integral

$\int x\cos x \, dx.\!$

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

$x\sin x - \int 1\sin x \, dx\!$

which equals

$x\sin x + \cos x+C.\!$

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 as dv, 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

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

one would set

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

so that

$du = 2x\,dx, \quad v = \frac12 e^{x^2}.$

Then

$\int x^3e^{x^2}\, dx = \int \left(x^2\right) \left( xe^{x^2} \right) \, dx = \int u \, dv = uv - \int v\,du = \frac12 x^2 e^{x^2} - \int xe^{x^2}\,dx.$

Finally, this results in

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

Recursive integration by parts

Integration by parts can often be applied recursively on the $\int v\,du$ term to provide the following formula

$\int uv = u v_1 - u' v_2 + u'' v_3 - \cdots + (-1)^{n}\ u^{(n)} \ v_{n+1}.\!$

Here, $u'\!$ is the first derivative of $u\!$ and $u''\!$ is the second derivative. Further, $u^{(n)}\!$ is a notation to describe its nth derivative with respect to the independent variable. Another notation approved in the calculus theory has been adopted:

$v_{n+1}(x)=\int\! \int\ \cdots \int v \ (dx)^{n+1}.\!$

There are n + 1 integrals.

Note that the integrand above (uv) differs from the previous equation. The dv factor has been written as v purely for convenience.

The above mentioned form is convenient because it can be evaluated by differentiating the first term and integrating the second (with a sign reversal each time), starting out with uv₁. It is very useful especially in cases when u^(k+1) becomes zero for some k + 1. Hence, the integral evaluation can stop once the u^(k) term has been reached.

Tabular integration by parts

While the aforementioned recursive definition is correct, it is often tedious to remember and implement. A much easier visual representation of this process is often taught to students and is dubbed either "the tabular method",^[2] "the Stand and Deliver method",^[3] "rapid repeated integration" or "the tic-tac-toe method". This method works best when one of the two functions in the product is a polynomial, that is, after differentiating it several times one obtains zero. It may also be extended to work for functions that will repeat themselves.

For example, consider the integral

$\int x^3 \cos x \, dx.\!$

Let u = x³. Begin with this function and list in a column all the subsequent derivatives until zero is reached. Secondly, begin with the function v (in this case cos(x)) and list each integral of v until the size of the column is the same as that of u. The result should appear as follows.

Derivatives of u (Column A)	Integrals of v (Column B)
$x^3 \,$	$\cos x \,$
$3x^2 \,$	$\sin x \,$
$6x \,$	$-\cos x \,$
$6 \,$	$-\sin x \,$
$0 \,$	$\cos x \,$

Now simply pair 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 alternating signs (beginning with the positive sign). Do so until further pairing is impossible. The result is the following (notice the alternating signs in each term):

$(+)(x^3)(\sin x) + (-)(3x^2)(-\cos x) + (+)(6x)(-\sin x) + (-)(6)(\cos x) + C \,.$

Which, with simplification, leads to the result

$x^3\sin x + 3x^2\cos x - 6x\sin x - 6\cos x + C. \,$

With proper understanding of the tabular method, it can be extended. Consider

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

Derivatives of u (Column A)	Integrals of v (Column B)
$e^x \,$	$\cos x \,$
$e^x \,$	$\sin x \,$
$e^x \,$	$-\cos x \,$

In this case in the last step it is necessary to integrate the product of the two bottom cells obtaining:

$\int e^x \cos x \,dx = e^x\sin x + e^x\cos x - \int e^x \cos x \,dx,$

which leads to

$2 \, \int e^x \cos x \,dx = e^x\sin x + e^x\cos x,$

and yields the result:

$\int e^x \cos x \,dx = {e^x ( \sin x + \cos x ) \over 2} + C\!$

Higher dimensions

The formula for integration by parts can be extended to functions of several variables. Instead of an interval one needs to integrate over an n-dimensional set. Also, one replaces the derivative with a partial derivative.

More specifically, suppose Ω is an open bounded subset of $\mathbb{R}^n$ with a piecewise smooth boundary Γ. If u and v are two continuously differentiable functions on the closure of Ω, then the formula for integration by parts is

$\int_{\Omega} \frac{\partial u}{\partial x_i} v \,d\Omega = \int_{\Gamma} u v \, \nu_i \,d\Gamma - \int_{\Omega} u \frac{\partial v}{\partial x_i} \, d\Omega$

where $\hat{\mathbf{\nu}}$ is the outward unit surface normal to Γ, $\mathbf{\nu}_i$ is its i-th component, and i ranges from 1 to n.

We can obtain a more general form of the integration by parts by replacing v in the above formula with v_i and summing over i gives the vector formula

$\int_{\Omega} \nabla u \cdot \mathbf{v}\, d\Omega = \int_{\Gamma} (u\, \mathbf{v})\cdot \hat{\nu}\, d\Gamma - \int_\Omega u\, \nabla\cdot\mathbf{v}\, d\Omega$

where v is a vector-valued function with components v₁, ..., v_n.

Setting u equal to the constant function 1 in the above formula gives the divergence theorem. For $\mathbf{v}=\nabla v$ where $v\in C^2(\bar{\Omega})$, one gets

$\int_{\Omega} \nabla u \cdot \nabla v\, d\Omega = \int_{\Gamma} u\, \nabla v\cdot\hat{\nu}\, d\Gamma - \int_\Omega u\, \Delta v\, d\Omega$

which is the first Green's identity.

The regularity requirements of the theorem can be relaxed. For instance, the boundary Γ need only be Lipschitz continuous. In the first formula above, only $u,v\in H^1(\Omega)$ is necessary (where H¹ is a Sobolev space); the other formulas have similarly relaxed requirements.

See also

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

Notes

1. ^ Kasube, Herbert E. (1983). "A Technique for Integration by Parts". The American Mathematical Monthly 90 (3): 210–211. doi:10.2307/2975556. JSTOR 2975556. 
2. ^ Khattri, Sanjay K. (2008). "FOURIER SERIES AND LAPLACE TRANSFORM THROUGH TABULAR INTEGRATION". The Teaching of Mathematics XI (2): 97–103. http://elib.mi.sanu.ac.rs/files/journals/tm/21/tm1125.pdf. 
3. ^ Horowitz, David (1990). "Tabular Integration by Parts". The College Mathematics Journal 21 (4): 307–311. doi:10.2307/2686368. JSTOR 2686368. http://www.maa.org/pubs/Calc_articles/ma035.pdf. 

References

• Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2. 
• Arbogast, Todd; Jerry Bona (2005) (PDF). Methods of Applied Mathematics. http://www.math.utexas.edu/users/arbogast/appMath08.pdf. 
• Horowitz, David (September 1990). "Tabular Integration by Parts". The College Mathematics Journal 21 (4): 307–311. doi:10.2307/2686368. JSTOR 2686368. 

External links

• Integration by Parts – From MathWorld
• Methods of integration -- section from an online textbook
• Tabular Integration by Parts
• Tabular Integration by Parts Demonstrated