Find : ∫dx/sin x + sin 2x

Question

Find : \(\int\frac{dx}{sin\,x+sin\,2x}dx\)

Answer 1

Here,

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

Here, integrand is proper rational function.

Therefore, by the form of partial function, we can write,

Again, putting the value of z = 1 in (ii) we get

⇒ 1 = 0 + B.2.(1+2) + 0

⇒ 1 = 6B

⇒ B = \(\frac{1}{2}\)

Similarly, putting the value of z = \(-\frac{1}{2}\) in (ii), we get

⇒ 1 = 0 + 0 + C \((\frac{1}{2})\)\((\frac{3}{2})\)

⇒ 1 = \((\frac{3}{4})C\)

⇒ C = \(\frac{4}{3}\) 

Putting the value of A, B, C in (i), we get

Putting the value of z, we get