Question

\[\int\text{ log }\left( x + 1 \right) \text{ dx }\]

Answer 1

\[\int \text{ log } \left( x + 1 \right)dx\]
\[ = \int1 . \log \left( x + 1 \right)dx\]
\[\text{Taking log} \left( x + 1 \right) \text{ as the first function and 1 as the second function} . \]
\[ = \text{ log }\left( x + 1 \right)\int \text{ 1 dx } - \int\left[ \frac{d}{dx}\left\{ \log\left( x + 1 \right) \right\}\int1 dx \right]dx\]
\[ = x \text{ log} \left( x + 1 \right) - \int\frac{x}{x + 1}dx\]
\[ = x \text{ log }\left( x + 1 \right) - \int\frac{x + 1}{x + 1} - \frac{1}{x + 1}dx\]
\[ = x \text{  log }\left( x + 1 \right) - x + \text{ log } \left| x + 1 \right| + C\]