Mathematics - Indefinite Integration Question with Solution | TestHub

$\int \frac{e^x(1+x)}{{\cos }^2\left(e^{x}x\right)}\mathrm{dx}$ Let $x{e}^x=t$ $\Rightarrow \left(x{e}^x+e^x\right)=\frac{\mathrm{dt}}{\mathrm{dx}}$ $\Rightarrow dx=\frac{\mathrm{dt}}{e^x(x+1)}$ $\therefore \int \frac{e^x(1+x)}{{\cos }^2\left(e^{x}x\right)}\mathrm{dx}=\int \frac{e^x(1+x)}{{\cos }^{2}t}\times \frac{dt}{e^x(1+x)}=\int \frac{1}{{\cos }^{2}t}\mathrm{dt}=\int {\sec }^{2}t\mathrm{dt}=$$\tan t+\mathrm{C}=\tan \left(x{e}^x\right)+\mathrm{C}$