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:
Rearranging, we have:
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)`
`intx\ sin\ 2x\ dx`
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`
Substituting these 4 expressions into the integration by parts formula, we get:
`intx^2\ ln\ 4x\ dx`
`intx\ sec^2\ x\ dx`
`intln\ x\ dx`
`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.