Question

Evaluate: ∫ (x²/ (x sin x + cos x)²)  dx 

Answer 1

∫ (x²/ (x sin x + cos x)² ) dx 

= ∫ (x cos x/ x sin x + cos x) . x sec x dx 

= x sec x ∫ (x cos x/ x sin x + cos x) dx – ∫ ( – (sec x + x sec x tan x)/ x sin x + cos x) dx 

= (x sec x/ x sin x + cos x)  + ∫ sec x dx + c 

=  (x sec x/ x sin x + cos x)  + log |sec x + tan x| + c