Question

integration of tan2x tan3x tan5x

Answer 1

Noticing that tan(5x) = tan(3x+2x) we use the tan compound angle formula to find tan
(5x) = (tan(2x)+tan(3x))/(1-tan(2x)tan(3x))
and thus
tan(5x)tan(3x)tan(2x) = tan(5x)-tan(3x)-tan(2x). From then we can integrate the parts of the sum individually as normal. Remembering that if F(x) = integral of f(x) dx then the integral of f(ax) dx = 1/a F(ax)

Answer 2

Answer:

Step-by-step explanation:

Let tn=tan(nx), we have

t5=t3+t21−t3t2⟺t5−t5t3t2=t3+t2⟹t5t3t2=t5−t3−t2

This leads to

∫tan(5x)tan(3x)tan(2x)dx=∫(tan(5x)−tan(3x)−tan(2x))dx=12logcos(2x)+13logcos(3x)−15logcos(5x)+ const.