# 7. Integration by Parts

by M. Bourne

Sometimes we meet an integration that is the product of 2 functions. We may be able to integrate such products by using Integration by Parts.

If u and v are functions of x, the product rule for differentiation that we met earlier gives us:

d/(dx)(uv)=u(dv)/(dx)+v(du)/(dx)

Rearranging, we have:

u(dv)/(dx)=d/(dx)(uv)-v(du)/(dx)

Integrating throughout with respect to x, we obtain the formula for integration by parts:

intu\ dv=uv-intv\ du

This formula allows us to turn a complicated integral into more simple ones. We must make sure we choose u and dv carefully.

NOTE: The function u is chosen so that (du)/(dx) is simpler than u.

## Priorities for Choosingu

When you have a mix of functions in the expression to be integrated, use the following for your choice of u, in order.

1. Let u = ln\ x

2. Let u = x^n

3. Let u = e^(nx)

### Example 1

intx\ sin\ 2x\ dx

#### Solution

We need to choose u. In this question we don't have any of the functions suggested in the "priorities" list above.

We could let u = x or u = sin\ 2x. In general, we choose the one that allows (du)/(dx) to be of a simpler form than u.

So for this example, we choose u = x and so dv will be the "rest" of the integral, dv = sin 2x dx.

We have u = x so du = dx.

Also dv = sin\ 2x\ dx and integrating gives:

v=intsin\ 2x\ dx

=(-cos\ 2x)/2

Substituting these 4 expressions into the integration by parts formula, we get:

### Example 2

intxsqrt(x+1)\ dx

### Example 3

intx^2\ ln\ 4x\ dx

### Example 4

intx\ sec^2\ x\ dx

### Example 5

intx^2e^(-x)dx

### Example 6

intln\ x\ dx

### Example 7

intarcsin\ x\ dx

This time we integrated an inverse trigonometric function (as opposed to the earlier type where we obtained inverse trigonometric functions in our answer). See Integration: Inverse Trigonometric Forms.

### Alternate Method for Integration by Parts

Here's an alternative method for problems that can be done using Integration by Parts. You may find it easier to follow.

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