# 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 Choosing**
**u**

**u**

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 2*x 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.

Didn't find what you are looking for on this page? Try **search**:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Go to: Online algebra solver

### Ready for a break?

Play a math game.

(Well, not really a math game, but each game was made using math...)

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

### Share IntMath!

### Short URL for this Page

Save typing! You can use this URL to reach this page:

intmath.com/intparts

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos