The formula of integration by parts states that

       (*)

for any differentiable functions  u (x)  and  v (x).

This formula allows to transform one problem of integration to another.

If one of the two integrals,    or  , is simple, it can be used to find the other one. This is the main idea of the method of integration by parts.

Formula (*) can be derived in the following way:

In practice, the procedure of integrating by parts consists of a few steps.

First, it is necessary to introduce intermediary functions  u (x)  and  v (x)  such that

.

For example, one can set  u (x) =f (x), which implies v'(x) = 1.

Next, one needs to differentiate  u (x)  and integrate  v'(x)  to get the differential

and the function

,

respectively.

Note that a constant of integration can be taken zero at this step (C = 0).

Then try to evaluate the integral  .

The main problem one faces when dealing with the method of integration by parts is the choice of the intermediary functions. There is no general rule to follow it. It is a matter of experience and nothing more. In order to understand better this technique, it is necessary to make any choice and perform the calculations. If the obtained integral is simpler than the given one, then all is O.K. Otherwise, go back and make another choice. In such a way one can easily appreciate whether the choice of  u (x)  is the best.

1. The integral of  v'  should be easy for evaluation.
2. The derivative of  u (x)  should be a simple function than  u (x).

Indefinite Integrals Definite Integrals Differential Equations