Question

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

Answer 1

$I=\int \frac{e^x(1+x))}{{\cos }^2(e^x\cdot x)}dx$ (1)

Let $e^x\cdot x=t$.

Differentiate above equation with respect to $x$.

$\frac{d}{dx}\left(e^{x}x\right)=\frac{dt}{dx}$

$e^x+x{e}^x=\frac{dt}{dx}$

$e^x(x+1)dx=dt$

Substitute $e^{x}x=t$ and $e^x(x+1)dx=dt$ in equation (1).

$I=\int \frac{dt}{{\cos }^{2}t}$

$={\sec }^{2}tdt$

$=\tan t+C$ (2)

Here, $C$ is integration constant.

Substitute $t=e^{x}x$ in equation (2).

$I=\tan \left(e^{x}x\right)+C$

Thus, the integration of $I=\int \frac{e^x(1+x)}{{\cos }^2(e^x\cdot x)}dx$ is $I=\tan \left(e^{x}x\right)+C$, where $C$ is integration constant.