Indefinite Integration By Parts

One reason it is harder to integrate than differentiate is that for derivatives there is both a Sum Rule and a Product Rule,

d(u + v) = du + dv, d(uv) = u dv + v du

while for integrals there is only a Sum Rule,

∫du + dv = ∫du + ∫dv.

The Sum Rule for integrals is obtained in a simple way by reversing the sum rule for derivatives.

There is a way to turn the Product Rule for derivatives into a rule for integrals. It no longer looks like a product rule, and is called integration by parts. Integration by parts is a basic method which is needed for many integrals involving trigonometric functions (and later exponential functions).

INDEFINITE INTEGRATION BY PARTS

Suppose, for x in an open interval I, that u and v depend on x and that du and dv exist. Then

∫u dv = uv - ∫v du.

No constant of integration is needed because there are indefinite integrals on both sides of the equation.

Integration by parts is useful whenever ∫ v du is easier to evaluate than a given integral ∫ u dv.