Solution of symbolic integration in Wolfram-Mathematica

Question

I am trying to get the solution of the following symbolic integral using Wolfram-Mathematica:

Integrate[1/(w^2 + l^2 + 2*l*w*Sin[t*w]), w]

but it does not return any solutions. Any ideas how to fix this?

Answer 1

When you encounter trigonometric functions (sin, cos), it is often useful to go to the exponential form (see the Help on TrigToExp).

This solves your integral easily:

solutionExp = Integrate[TrigToExp[w^2 + l^2 + 2 l w Sin[t w]], w]

This solution can be brought back to trigonometric form with ExpToTrig:

solutionTrig = ExpToTrig[solutionExp]