Question

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

Answer 1

\[\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x \cdot e^x \right)} dx\]
\[\text{Let x e}^x = t\]
\[ \Rightarrow \left( 1 \cdot e^x + \text{x  e}^x \right) = \frac{dt}{dx}\]
\[ \Rightarrow \left( x + 1 \right) e^x dx = dt\]
\[Now, \int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x \cdot e^x \right)} dx\]
\[ = \int\frac{dt}{\cos^2 t}\]
\[ = \int \sec^2 \text{t dt}\]
\[ = \tan \left( t \right) + C\]
` = tan   ( x  e^x)   + C `