This section introduces one of the general techniques of integration, integration by parts.

If function u = u(x) and v = v(x) are differentiable functions, according to the Produce Rule, the differentiation of the product of functions u and v is

      (uv) ^' = u ^'v + uv ^'

Rearrange the above equation gives

      uv ^' = (uv) ^' - u ^'v

Integrating both sides (indefinite integrals),

Since v^' dx = dv, and u^' dx = du, for simple, the formula for integration by parts becomes

Integration xsinx using Integration by Parts (Definite Integration)

Evaluating both sides of the previous equation between a and b, assuming u ^' and v ^' are continuous, becomes,

      

If udv is difficult to integrate and vdu is easy, then integration by parts is useful.

The key in using integration by parts is how to choose u and dv. Two things should be considered,

• v is easy to integrate from dv, and
• compare with , is easier to integrate.