Q.377. Indefinite integration.

Question 377.
Find ∫ [(1 + sinx) / (1 + cosx)] e^x dx.

Answer 377.
e^x * [(1 + sinx) / (1 + cosx)]
= e^x * [1 + 2sin(x/2) cos(x/2)] / [2cos^2 (x/2)]
= [(1/2) sec^ (x/2) + tan(x/2)] e^x

 Let f (x) = tan(x/2)
=> f '(x) = (1/2) sec^2 (x/2)

 => ∫ e^x [(1+sinx)/(1 + cosx)] dx
= ∫ [(1/2) sec^ (x/2) + tan(x/2)] e^x dx
= ∫ [ f (x) + f '(x) ] e^x dx
= f (x) e^x + c
= tan(x/2) e^x + c.

Link to YA!