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:
Rearranging, we have:
Integrating throughout, with respect to x, we obtain the formula for integration by parts:
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 
is simpler than
u.
Priorities for Choosing u
1. Let u = ln x
2. Let u = xn
3. Let u = enx
Example 1: 
Solution:
We could let u = x or u
= sin 2x. In general, we choose the one that allows
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
Substituting into the integration by parts formula, we get:
Example 2: 
Example 3: 
Example 4: 
Example 5: 
Example 6: 
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