JEE Advance - Mathematics (1996 - No. 33)
Evaluate $$\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx} $$.
$$\log \left( {\frac{{1 + x{e^x}}}{{x{e^x}}}} \right) - \frac{1}{{1 + x{e^x}}} + C$$
$$\log \left( {1 + x{e^x}} \right) - \frac{1}{{1 + x{e^x}}} + C$$
$$\log \left( {\frac{{1 + x{e^x}}}{x}} \right) - \frac{1}{{1 + x{e^x}}} + C$$
$$\log \left( {\frac{{x{e^x}}}{{1 + x{e^x}}}} \right) - \frac{1}{{1 + x{e^x}}} + C$$
$$\log \left( {\frac{{1 + x{e^x}}}{{x}}} \right) + \frac{1}{{1 + x{e^x}}} + C$$