Evaluate the following integrals : ∫(1+cotx)/(x+log sinx) dx ​

Question

Evaluate the following integrals :

\(\int\)\(\frac{1+cotx}{x+log\,sinx}\)dx

Answer 1

Assume,

x + log(sinx) = t 

d(x + log(sinx)) = dt 

1 + \(\frac{cos\,x}{sin\,x}\)dx = dt

(1 + cot)dx = dt 

Put t and dt in given equation we get,

⇒∫\(\frac{dt}{t}\)

= In|t| + c

But,

t = x + log(sinx) 

= ln| x + log(sinx) | + c.