An integral may be too complicated to evaluate completely by the exact methods of freshman calculus, such as substitution or integration by parts. Nevertheless, it is still worthwhile to use these methods as much as possible to simplify the integral to a more manageable form.

Note that "simpler" may not mean "cleaner" or "shorter". Sometimes, one would rather deal with a messy sum of ten easily controllable integrals than one short but difficult integral.

• Which integrals are simpler to integrate?

• Use rescaling or translation to normalize parameters

  • Select a favorable gauge

• Integration by parts

• Interchange integrals or sums

• Use an appropriate series expansion

• Adding and subtracting

• Use Fourier identities

• Smoothing sums

• Shift the contour of integration