Question

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

Answer 1

$$\begin{gathered} \int \frac{\left(x+1\right)e^x}{{\cos }^2\left(x·e^x\right)}dx \\ \mathrm{Let}x{e}^x=t \\ \Rightarrow \left(1·e^x+x{e}^x\right)=\frac{dt}{dx} \\ \Rightarrow \left(x+1\right)e^{x}dx=dt \\ \mathrm{Now},\int \frac{\left(x+1\right)e^x}{{\cos }^2\left(x·e^x\right)}dx \\ =\int \frac{dt}{{\cos }^{2}t} \\ =\int {\sec }^2\mathrm{t}\mathrm{dt} \\ =\tan \left(t\right)+C \\ =\tan \left(x{e}^x\right)+C \end{gathered}$$