Question

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

• A. $\tan \left(x{e}^x\right)+c$
• B. $\sec \left(x{e}^x\right)+c$
• C. $\sin \left(x{e}^x\right)+c$
• D. $\cot \left(x{e}^x\right)+c$

Answer 1

The correct option is A $\tan \left(x{e}^x\right)+c$

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

$=\int \frac{e^x(1+x)}{{\cos }^{2}t}\times \frac{dt}{e^x(1+x)}$

$=\int dt.{\sec }^{2}t$

$=\tan t+C$

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

$\Rightarrow$ Put $x{e}^x=t$

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

$e^x+e^{x}x=dt/dx$

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

$dk=\frac{dt}{(1+k)}{e}^k$