Q104 / 160 · AP EAMCET 2020 (22 Sep Shift 2) · Mathematics

$\int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot(1-x \cot x) d x=$

  • A
    $e^{x \cot x}+c$
  • Correct answer
    B
    $e^{x \operatorname{cosec} x}+c$
  • C
    $e^{-x \operatorname{cosec} x}+c$
  • D
    $e^{-x \cot x}+c$

Solution

$\int e^{x \operatorname{cosec}} x \operatorname{cosec} x(1-x \cot x) d x$ Put, $x \operatorname{cosec} x=t$ Differentiate w.r.to ' $x$ ' $$ \begin{aligned} & x \cdot \frac{d}{d x} \operatorname{cosec} x+\operatorname{cosec} x \cdot \frac{d}{d x} \cdot x=\frac{d}{d x} t \\ & x(-\operatorname{cosec} x \cdot \cot x)+\operatorname{cosec} x(1)=\frac{d t}{d x} \\ & \operatorname{cosec} x(1-x \cot x) d x=d t=\int e^t \cdot d t \\ &=e^t+c=e^{x \operatorname{cosec} x}+c \end{aligned} $$ Hence, option (2) is correct.

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

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