Q111 / 160 · TS EAMCET 2007 · Mathematics

If $\int \frac{e^x-1}{e^x+1} d x=f(x)+c$, then $f(x)$ is equal to

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

Solution

We have, $\int \frac{e^x-1}{e^x+1} d x$ $=\int\left(\frac{2 e^x}{e^x+1}-1\right) d x$ $=2 \log \left(e^x+1\right)-x+c$ On comparing with $f(x)+c$, we get $f(x)=2 \log \left(e^x+1\right)-x$

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

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