Q91 / 150 · MHT CET 2022 (07 Aug Shift 1) · Mathematics

$\int \frac{e^x}{\left(2+e^x\right)\left(e^x+1\right)} d x=$ (where $C$ is $a$ constant of integration.)

  • A
    $\frac{e^x+1}{e^x+2}+C$
  • B
    $\log \left(\frac{e^x+2}{e^x+1}\right)+C$
  • Correct answer
    C
    $\log \left(\frac{e^x+1}{e^x+2}\right)+C$
  • D
    $\log \left(\frac{e^x}{e^x+2}\right)+C$

Solution

$\begin{aligned} & \int \frac{e^x}{\left(2+e^x\right)\left(e^x+1\right)} d x \\ & =\int \frac{d t}{(t+2)(t+1)}=\int\left(\frac{-1}{t+2}+\frac{1}{t+1}\right) d t \\ & =-\log |t+2|+\log |t+1|+c \\ & =\log \left|\frac{t+1}{t+2}\right|+c \\ & =\log \left|\frac{e^x+1}{e^x+2}\right|+c\end{aligned}$

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

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