Question

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

• A. $\tan \left(x{e}^x\right)+c$
• B. ${\sec }^2\left(e{x}^x\right)+c$
• C. $\cos \left(x{e}^x\right)+c$
• D. $\cot \left(x{e}^x\right)+c$

Answer 1

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

Substitute $x{e}^x=t$

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

$\int \frac{dt}{co{s}^{2}t}=\int se{c}^{2}t\cdot dt$

$=\tan \left(t\right)+c$

$=\tan \left(x{e}^x\right)+c$

So , $\int \frac{e^x(1+x)}{co{s}^2\left(x{e}^x\right)}\cdot dx=\tan \left(x{e}^x\right)+c$