Evaluate the following integrals : ∫ cotx log sinx dx

1.	Evaluate the following integrals : ∫ cotx log sinx dx
Answer» Assume, log(sinx) = t d(log(sinx)) = dt ⇒ \(\frac{cosx}{sinx}\)dx = dt ⇒ cot x dx = dt Substituting the values oft and dt we get ⇒ \(\int\)t dt ⇒ \(\frac{t^2}{2}\) + c But, t = log(sinx) ⇒ \(\frac{log(sin\,x)^2x}{2}\) + c.

Answer»

Assume,

log(sinx) = t

d(log(sinx)) = dt

⇒ \(\frac{cosx}{sinx}\)dx = dt

⇒ cot x dx = dt

Substituting the values oft and dt we get

⇒ \(\int\)t dt

⇒ \(\frac{t^2}{2}\) + c

But,

t = log(sinx)

⇒ \(\frac{log(sin\,x)^2x}{2}\) + c.

Discussion