Q208 / 270 · NDA 2010 (Phase 2) · Mathematics

What is equal to? $\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x$ where $c$ is a constant of integration.

  • A
    $x e^x+c$
  • B
    $\cos \left(x e^x\right)+c$
  • Correct answer
    C
    $\tan \left(x e^x\right)+c$
  • D
    $x \operatorname{cosec}\left(x e^x\right)+c$

Solution

$\begin{aligned} & \int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x \\ & \text {Put } x e^x=z \\ & \therefore e^x(1+x) d x=d z \\ & \int \frac{d z}{\cos ^2 z}=\int \sec ^2 z d z=\tan z+c \\ & =\tan \left(x e^x\right)+c\end{aligned}$

Hence, the correct answer is option: 'C'.

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