Q120 / 160 · AP EAMCET 2020 (22 Sep Shift 2) · Mathematics

$\int \frac{(1+x) e^x}{\cot \left(x e^x\right)} d x=$

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

Solution

$\int \frac{(1+x) \cdot e^x}{\cot \left(x \cdot e^x\right)} d x$ Put, $x e^x=t$ Differentiate w.r.to, $x$ $$ \begin{aligned} \Rightarrow e^x(1+x) d x & =d t \\ & =\int \frac{d t}{\cot t}=\int \tan t d t \\ & =\log (\sec t)+c \\ & =\log \left(\sec \left(x e^x\right)\right)+c \end{aligned} $$ Hence option (3) is correct.

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

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