Question

\[\int\frac{e^x \left( 1 + x \right)}{\cos^2 \left( x e^x \right)} dx =\]

Options

• 2 log_e cos (xe^x) + C

• sec (xe^x) + C

• tan (xe^x) + C

•  tan (x + e^x) + C

Answer 1

tan (xe^x) + C

 

\[\text{Let }I = \int\frac{e^x \left( 1 + x \right)}{\cos^2 \left( x e^x \right)}dx\]
\[\text{Putting }x e^x = t\]
\[ \Rightarrow \left( 1 \cdot e^x + x e^x \right)dx = dt\]
\[ \Rightarrow e^x \left( 1 + x \right)dx = dt\]
\[ \therefore I = \int\frac{dt}{\cos^2 t}\]
\[ = \int \sec^2 t dt\]
\[ = \tan t + C\]
\[ = \tan \left( x e^x \right) + C ............\left( \because t = x e^x \right)\]