Integration by parts

Integration by parts (IBP) is a method of integration with the formula:


Or more compactly,

      or without bounds      

where and are functions of a variable, for instance, , giving and .

      and      

Note: is whatever terms are not included as .

LIATE RuleEdit

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

L – logarithmic functions: etc.
I – inverse trigonometric functions: etc.
A – polynomials: etc.
T – trigonometric functions: etc.
E – exponential functions: etc.

DerivationEdit

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

 

Integrating both sides with respect to  ,

 

and noting that an indefinite integral is an antiderivative gives

 

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

 

or in terms of the differentials  

 

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:

 

ExamplesEdit

Functions multiplied by one and itselfEdit

Given  Edit

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

 

Let:

 
 

then:

 

where C is the constant of integration.

Given  Edit

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

 

Rewrite this as

 

Now let:

 
 

then

 

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

Polynomials and trigonometric functionsEdit

In order to calculate

 

let:

 
 

then:

 

where C is a constant of integration.

For higher powers of x in the form

 

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

Exception to LIATEEdit

 

one would set

 

so that

 

Then

 

Finally, this results in

 

Performing IBP twiceEdit

 

Here, integration by parts is performed twice. First let

 
 

then:

 

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

 
 

Then:

 

Putting these together,

 

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

 

which rearranges to