Evaluate: ∫dx/sinx+sin 2x

Question

Evaluate: ∫\(\frac{dx}{sinx+sin\ 2x}\)

Answer 1

Let

[Multiplying numerator and denominator by sin x]

= ∫\(\frac{sin\ x}{(1-cos^2x)(1+2cos\ x)}\) dx

Put cos x = t

⇒ - sin x dx = dt

⇒ sin x dx = - dt

∴ I = ∫\(\frac{-dt}{(1-t^2)(1+2t)}\) 

 = ∫\(\frac{-dt}{(1-t)(1+t)(1+2t)}\) ...... (i)

Now, using partial fraction,

Let \(\frac{1}{(1-t)(1+t)(1+2t)} = \frac{A}{1-t} + \frac{B}{1+t} + \frac{C}{1+2t}\) ...... (ii)

⇒ 1 = (1 + t) (1 + 2t)A + (1 - t) (1 + 2t)B + (1 - t) (1 + t)C ......... (iii)

On putting t = - 1 in Eq.(iii), we get

1 + (2) (- 1) B ⇒ B = \(-\frac{1}{2}\)

On putting t = 1 in Eq. (lu), we get

1 = 2. (3) A ⇒ A = \(-\frac{1}{6}\)

On putting t = \(-\frac{1}{2}\) in Eq. (iii), we get