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Question: Answered & Verified by Expert
$\int \frac{d x}{1+e^x}=$
MathematicsIndefinite IntegrationJEE Main
Options:
  • A $\log \left(1+e^x\right)$
  • B $-\log \left(1+e^{-x}\right)$
  • C $-\log \left(1-e^{-x}\right)$
  • D $\log \left(e^{-x}+e^{-2 x}\right)$
Solution:
1960 Upvotes Verified Answer
The correct answer is: $-\log \left(1+e^{-x}\right)$
$\int \frac{d x}{1+e^x}=\int \frac{e^{-x}}{1+e^{-x}} dx$
Put $1+e^{-x}=t \Rightarrow e^{-x} d x=-d t$, then it reduces to $-\int \frac{d t}{t}=-\log t=-\log \left(1+e^{-x}\right)$

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