Question

Evaluate: ∫ (sec x (2 + sec x)/ (1 + 2sec x)²) dx

Answer 1

The given integral is

∫ (sec x (2 + sec x)/ (1 + 2sec x)²) dx 

= ∫ (2cos x + 1/ (cos x + 2)²) dx

= ∫ ((cos x (cos x + 2) + sin x)/ (cos x + 2)²) dx

= ∫ (cos x/ cos x + 2) dx + ∫ (sin x (cos x + 2)²) dx

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

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