In mathematics, **integration by parts** is a general rule that transforms the integral of calculus of products of functions into other integrals. The objective is that these are simpler. The rule arises from the product rule of differentiation.

Suppose *f*(*x*) and *g*(*x*) are two continuously differentiable functions. Then the rule states

*u*=

*f*(

*x*),

*v*=

*g*(

*x*) and the differentials

*du*=

*f*′(

*x*)

*dx*and

*dv*=

*g*′(

*x*)

*dx*, then it is in the form in which it is most often seen:

Note that the original integral contains the derivative of *g*; in order to be able to apply the rule, you need to find its antiderivative *g* and then you still have to evaluate the resulting integral of ∫*g* *f* ' d*x*.

An alternative notation has the advantage that the factors of the original expression are identified as *f* and *g*, but the drawback of a nested integral:

*f*is continuously differentiable and

*g*is continuous.

If we combine the first formula above with the fundamental theorem of calculus, definite integrals can also be integrated by parts. If we evaluate both sides of the formula between *a* and *b* and assume *f(x)* and *g(x)* are continuous, by applying the Fundamental Theorem of Calculus, we obtain this useful formula:

Table of contents |

2 Examples 3 Justification of the rule 4 Connection to distributions |

## Application

## Examples

Let:

*u*=*x\*, so that*du*=*dx*,*dv*= cos(*x*)*dx*, so that*v*= sin(*x*).

*C*is an arbitrary constant of integration.

By repeatedly using integration by parts, integrals such as

*x*by one.

An interesting example that is commonly seen is:

This example uses integration by parts twice. First let:

*u*= e^{x}; thus d*u*= e^{x}d*x**v*= sin(*x*); thus d*v*= cos(*x*)d*x*

*u*= e^{x}; d*u*= e^{x}d*x**v*= -cos(*x*); d*v*= sin(*x*)d*x*

*x*.

The first example is ∫ ln(*x*) d*x*. Write this as:

*u*= ln(*x*); d*u*= 1/*x*d*x**v*=*x*; d*v*= 1·d*x*

The second example is ∫ arctan(*x*) d*x*, where arctan(*x*) is the inverse tangent function. Re-write this as:

*u*= arctan(*x*); d*u*= 1/(1+*x*^{2}) d*x**v*=*x*; d*v*= 1·d*x*

## Justification of the rule

Integration by parts follows from the product rule of differentiation: If the two continuously differentiable functions *u*(*x*) and *v*(*x*) are given, the product rule states that

*u*and

*v*are

*continuously*differentiable ensures that the two individual integrals exist.) Subtracting ∫

*u*

*v*' d

*x*from both sides yields the desired formula of integration by parts.

## Connection to distributions

When defining distributions, integration rather then differentiation is the fundamental operation. The derivatives of distributions are then *defined* so as to make integration by parts work.