7. Integration by Parts

by M. Bourne

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

math expression

Rearranging, we have:

math expression

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

math expression

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

Function u is chosen so that math expression is simpler than u.

Priorities for Choosing u

1. Let u = ln x

2. Let u = xn

3. Let u = enx

Example 1: math expression

Solution:

We could let u = x or u = sin 2x. In general, we choose the one that allows math expression to be of a simpler form than u.

So for this example, we choose u = x and dv = sin 2x dx.

u = x
 dv = sin 2x dx
du = dx math expression

Substituting into the integration by parts formula, we get:

math expression


Example 2: math expression

Example 3: math expression

Example 4: math expression

Example 5: math expression

Example 6: math expression

Example 7: \large{\int{\arcsin{x}\hspace{2}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 easier to follow.

Tanzalin Method

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