Question

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

(a) 2 log_e cos (xe^x) + C

(b) sec (xe^x) + C

(c) tan (xe^x) + C

(d) tan (x + e^x) + C

Answer 1

(c) tan (xe^x) + C

$$\begin{gathered} \mathrm{Let}I=\int \frac{e^x\left(1+x\right)}{{\cos }^2\left(x{e}^x\right)}dx \\ \mathrm{Putting}x{e}^x=t \\ \Rightarrow \left(1·e^x+x{e}^x\right)dx=dt \\ \Rightarrow e^x\left(1+x\right)dx=dt \\ \therefore I=\int \frac{dt}{{\cos }^{2}t} \\ =\int {\sec }^{2}tdt \\ =\tan t+C \\ =\tan \left(x{e}^x\right)+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\mathit{}t=\mathit{}x{e}^x\right) \end{gathered}$$