Start: Identify integral type
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Is integrand a rational function P$\frac{x}{Q}$(x)?
  Yes -> Partial fractions
  No -> Continue
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Does it contain sqrt(linear)?
  Yes -> Substitute $t^2$ = linear expression
  No -> Continue
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Does it contain sqrt(quadratic)?
  Yes -> Complete square, then trig substitution
  No -> Continue
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Is it a product of two function types?
  Yes -> Integration by parts (LIATE rule)
  No -> Continue
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Is it $e^x$ * [f(x) + f'(x)]?
  Yes -> Answer is $e^x$ * f(x) + C
  No -> Continue
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Is it rational in sin(x), cos(x)?
  Yes -> Check symmetry, choose substitution
  No -> Continue
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Is it a power of trig function?
  Yes -> Use reduction formula or power-lowering identities
  No -> Try algebraic manipulation or Weierstrass