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GATE 2017 CE (CE1) – Question 13

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 1 mark · Multiple choice

For $x\in(-\infty,\infty)$, let $f(x)=e^{-x-e^{-x}}$. The indefinite integral $g(x)=\int f(x)\,dx$ is equal to (up to an additive constant)

  1. $e^{e^{-x}}$
  2. $e^{-e^{-x}}$
  3. $e^{-e^x}$
  4. $e^{-x}$

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Correct answer: (B) $e^{-e^{-x}}$

Explanation

Put $u=e^{-x}$, so $du=-e^{-x}dx$. Then $\int e^{-x}e^{-e^{-x}}dx=-\int e^{-u}du=e^{-u}+C=e^{-e^{-x}}+C$. Differentiating the expression in B gives $e^{-e^{-x}}e^{-x}=f(x)$, confirming the sign.

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