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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2010 · Mathematics · Indefinite Integration

What is ∫ (1)/(1+e^x) d x equal to? where c is constant of...

What is ∫11+exdx\int \frac{1}{1+e^x} d x equal to? where cc is constant of integration.

Options

  1. Ax−ln⁡x+cx-\ln x+c
  2. Bx−ln⁡(tan⁡x)+cx-\ln (\tan x)+c
  3. Cx−ln⁡(1+ex)+cx-\ln \left(1+e^x\right)+c
  4. Dln⁡(1+ex)+c\ln \left(1+e^x\right)+c

Correct answer

C. x−ln⁡(1+ex)+cx-\ln \left(1+e^x\right)+c

Step-by-step solution

∫11+exdx=∫1+ex−ex1+exdx=∫1+ex1+exdx−∫ex1+exdx=x−log⁡(1+ex)+c\begin{aligned} & \int \frac{1}{1+e^x} d x \\ & =\int \frac{1+e^x-e^x}{1+e^x} d x \\ & =\int \frac{1+e^x}{1+e^x} d x-\int \frac{e^x}{1+e^x} d x \\ & =x-\log \left(1+e^x\right)+\mathrm{c}\end{aligned}
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