Question

Evaluate: ∫ (((cos x – sin x)/ (cos x + sinx)) × (2 + 2sin 2 x)) dx

Answer 1

∫ ((cos x – sin x)/ (cos x + sin x)× (2 + 2sin 2x) dx 

= ∫ (((cos x – sinx)/ (cos x + sinx )) (1 + (sinx + cos x)²)) dx 

= ∫ (1 + t²/ t) dt, Let cos x + sin x = t 

= (– sin x + cos x) dx = dt 

= ∫ (t + (1/t)) dt 

= (t²/2) + log |t| + c 

= ((sin x + cos x)²/ 2) + log |sin x + cos x| + c