Integration by partsis one of the powerful techniques for solving integrals involving products of functions. We apply this technique whenever the U-substitution fails to solve such products.By playing some tricks with the product rule for derivatives,

We obtain the integration by parts formula;

There are some combinations that are very tricky to solve. These include :

Let’s consider how to evaluate the first two then you try the last one yourself.

Evaluate:

##### solution

Let

and

This is a u-substitution on its own. Since we used u already, let’s make it a w-substitution

This is another *by-parts* on its own. Here is its value:

Back to main work:

Let’s verify our solution by evaluating the definite integral:

Now let’s continue with the second:

Evaluate:

Solution

And

But