Integration by parts is a technique for evaluating indefinite integral whose integrand is a product of two functions. It is based on the fact that

If two functions are differentiable on an interval and exists, then

It is not difficult to see that (1) is true:

Provide exist, let

and

.

By the rules of differentiation (see “Some rules of differentiation”),

.

i.e.,

or,

The key to apply this technique successfully is to choose proper so that the integrand in the original integral can be expressed as and, is easier to evaluate than .

To evaluate , we let to get

In fact, for ,

.

The following example of integrating a rather ordinary-looking expression offers unexpected difficulties and surprises:

Notice the original integral (boxed) appears on the right of the “=” sign. However, this is not an indication that we have reached a dead end. To the contrary, after it is combined with the left side, we have

and so,

Sometimes, successive integration by parts is required to complete the integration. For example,

.

Hence,

i.e.,

Exercise-1 Evaluate

1)

2)

3)

While Maxima choked:

Mathematica came through:

4)

5)

6)

7)

Both Maxima and Mathematica came though: