Question

Write a value of

\[\int\frac{1 + \log x}{3 + x \log x} \text{ dx }\] .

Answer 1

\[\text{ Let I } = \int \left( \frac{1 + \log x}{3 + x \log x} \right)dx\]
\[\text{ Let 3 }+ x \log x = t\]
\[ \Rightarrow 0 + \left( x . \frac{1}{x} + \log x \right)dx = dt\]
\[ \Rightarrow \left( 1 + \log x \right)dx = dt\]
\[ \therefore I = \int \frac{dt}{t}\]
\[ = \text{ log t + C }\]
\[ = \text{ log }\left( 3 + x \log x \right) + C \left( \because t = 3 + x \log x \right)\]