Question

Integral of cos2x log(3+4tanx)dx

Answer 1

The integral is:- sinxcosxlog(3+4tanx) +1/25(12log(4sinx+3cosx)+16tanx)+C

1. Integrate by using Integration by Part  ∫ f g dx = f ∫gdx − ∫ f ′( ∫gdx)dx , first integrate cos(2x) and differentiate log(3+4tanx) (i.e. let f= log(3+4tanx) and g=cos(2x))
2. Then substitute tanx = y
3. and then use Partial fraction method.
4. solve the simpler integrals by substitution or direct formula.

Please refer to the attachment.