Evaluate: ∫dx/(sin x + sin 2x) ​

Question

Evaluate:

\(\int\frac{dx}{(sin\,x\,+sin2\,x)}\)

∫dx/(sin x + sin 2x)

Answer 1

Let I = \(\int\frac{dx}{(sin\,x\,+sin2\,x)}\) = \(\int\frac{dx}{(sin\,x\,+2sin\,x\,cos\,x)}\)

Put t = cos x 

dt = - sinxdx

A(1 + t)(1 + 2t) + B(1 - t)(1 + 2t) + C(1 - t2) = 1 

Putting 1 + t = 0 

t = - 1 

A(0) + B(2)(1 - 2) + C(0) = 1

B = - 1/2

Putting 1-t = 0 

t =1 

A(2)(3) + B(0) + C(0) = 1

A = 1/6

Putting 1 + 2t = 0

t = - 1/2