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


Answer

Example 3: math expression


Answer

Example 4: math expression


Answer

Example 5: math expression


Answer

Example 6: math expression


Answer



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