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

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:

Integration by parts example steps

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.

Tanzalin Method

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