Previous page : Integration of definite integrals using substitution

Next page : One function is in the form x to the power of n and the other can be integrated at least n times

**Integration by parts**

In the case the integrand is a product of two functions we may be able to solve it by substitution, as shown in the previous pages. But what if does not work? One way one could try is a kind of inverse of the differential product rule. Say *u* and *v* are functions of *x*, then we have that

or

Integrating both sides gives us

or

This means that we can transform one integral into another that may be solvable. The goal is thus to find

**Example**: First an example to make some sense of this. Let us try to find

We cannot use the substitution method since none of the inner derivatives of the other function is the derivative of the other function. But let us instead set

and thus

That means that

We can test this by differentiating:

**Other ways to write the rule**

You might find this rule written as

Here the two functions have swapped order. Using this way of writing it might be a shortcut.

You might also see it written as

Here we have that

In the next few pages, we will look at some cases.

Up a level : IntegralsPrevious page : Integration of definite integrals using substitution

Next page : One function is in the form x to the power of n and the other can be integrated at least n timesLast modified: