Question

Write a value of

\[\int\frac{1 + \cot x}{x + \log \sin x} \text{ dx }\]

Answer 1

\[\text{ Let I }= \int\frac{1 + \cot x}{x + \text{ log  sin x}}dx\]
\[\text{ Let x } + \log \sin x = t\]
\[ \Rightarrow \left( 1 + \frac{1}{\sin x} \times \cos x \right) dx = dt\]
\[ \Rightarrow \left( 1 + \cot x \right)dx = dt\]
\[ \therefore I = \int\frac{dt}{t}\]
\[ = \text{ log }\left| t \right| + C\]
\[ = \text{ log } \left| x + \log \sin x \right| + C\]