Evaluate: ∫^(π/4)0 sinx+cosx/16+9sin 2x dx

Question

Evaluate: \(\int^{\pi/4}_0\) \(\frac{sinx+cosx}{16+9sin\ 2x}\) dx

Answer 1

Let I =  \(\int^{\pi/4}_0\) \(\frac{sinx+cosx}{16+9sin\ 2x}\) dx

[∵ a² + b² - 2ab = (a - b)²]

Put sin x - cos x = t ⇒ (cos x + sin x)dx = dt

Also, when, x = 0, then t = - 1

and when, x = \(\frac{\pi}{4}\) then t = 0