The integration by parts method is based on the product rule.
Example 1
Find:
A function is needed whose derivative is xex. Thinking about the product rule, one might guess xex to be the derivative. When the derivative is taken, two terms will be obtained, one of which will be xex:
| (xex ) = | | (x)ex + x | | (ex ) = ex + xex |
This guess is wrong because of the extra ex. Maybe the guess can be adjusted by subtracting from xex something to cancel out the extra ex. The integration of xex − ex will now be tried:
| (xex − ex ) = | | (xex ) − | | (ex ) = ex + xex − ex = xex |
It works, so:
Example 2
Find:
It may be guessed that the antiderivative is θ sin θ. The product rule is used to check:
| (θ sin θ ) = | | sin θ + θ | | (sin θ ) = sin θ + θ cos θ |
To correct for the extra sin θ term, something whose derivative is sin θ must be subtracted from the original guess. Because:
The following is tried:
| (θ sin θ + cos θ ) = | | (θ sin θ ) + | | (cos θ ) = sin θ + θ cos θ − sin θ = θ cos θ |
Therefore:
| ∫ | θ cos θ dθ = θ sin θ + cos θ + C |
General Forumula for Integration by Parts
The process followed in the previous examples can be formalized. Beginning with the product rule:
where u and v are functions of x with derivatives u' and v', respectively. This may be rewritten as:
Both sides are then integrated:
| ∫ | uv' dx = | ∫ | | (uv) dx − | ∫ | u' v dx |
Because an antiderivative of:
is just uv, the following formula, which represents the general method for integration by parts, results:
| (Eq1) | | ∫ | uv' dx = uv − | ∫ | u' v dx |
|
This formula is useful when the integrand can be viewed as a product and when the integral on the right-hand side is simpler than that on the left. In effect, integration by parts was being used in the previous two examples. In example 1:
xex = (x)(ex) = uv'
where u = x and v' = ex. Thus u' = 1 and v = ex, so: