2.5.1—Extended integration rules

Use substitution when an inner derivative is present, and integration by parts ∫u dv=uv−∫v du when a product becomes simpler after differentiating one factor.

For substitution, change the differential and limits consistently. For parts, choose u to become simpler and dv to be easy to integrate; differentiate the final answer to verify.

∫2x cos(x²)dx=sin(x²)+C by u=x². For ∫x e^x dx, take u=x and dv=e^x dx.

Substitution is not complete until every x and dx is replaced, and integration by parts does not remove the need for C in an indefinite integral.