Q91 / 150 · MHT CET 2020 (12 Oct Shift 1) · Mathematics

$\int_{-5}^{5}\left[\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\right] d x=$

  • Correct answer
    A
    0
  • B
    1
  • C
    $3 e^{5}$
  • D
    $2 e^{5}$

Solution

$\begin{aligned} \text { Let } f(x)=& \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} \\ =& \frac{e^{x}+\frac{1}{e^{x}}}{e^{x}-\frac{1}{e^{x}}}=\frac{e^{2 x}+1}{e^{2 x}-1} \text { and } f(-x)=\frac{e^{-x}+e^{x}}{e^{-x}-e^{x}}=\frac{\frac{1}{e^{x}}+e^{x}}{\frac{1}{e^{x}}-e^{x}}=\frac{1+e^{2 x}}{1-e^{2 x}} \\ \therefore f(x)=-f(x) \end{aligned}$ Thus $f(x)$ is an odd function. $\therefore \int_{-5}^{5} \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} d x=0$

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

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