Question

Evaluate $\int \frac{e^x(1+x)}{{\cos }^2\left(x{e}^x\right)}dx$ on $I\subset R$ \ $\{x\in R:\cos (x{e}^x)=0\}$

Answer 1

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

Let

$x{e}^x=u$

$(e^x+x{e}^x)dx=du$

$\Rightarrow e^x(1+x)dx=du$

Thus the integral will become-

$\int \frac{du}{{\cos }^{2}u}$

$=\int {\sec }^{2}udu$

$=\tan u+C$

Substituting the value of $u$ in above equation, we get

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

Hence

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