Integration by parts

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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 Rule[edit | edit source]

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.

Derivation[edit | edit source]

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:

Examples[edit | edit source]

Functions multiplied by one and itself[edit | edit source]

Given [edit | edit source]

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

Let:

then:

where C is the constant of integration.

Given [edit | edit source]

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 functions[edit | edit source]

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 LIATE[edit | edit source]

one would set

so that

Then

Finally, this results in

Performing IBP twice[edit | edit source]

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