GATE 2023 MN – Question 39

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Evaluate $\int\frac{e^{x-1}+x^{e-1}}{e^x+x^e}\,dx$. Each option is understood up to an additive constant.

$\ln(e^x+x^e)$
$e^{-1}\ln(e^x-x^e)$
$e^{-1}\ln(e^x+x^e)$
$\ln(e^x-x^e)$

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Correct answer: (C) $e^{-1}\ln(e^x+x^e)$

Explanation

Let g(x)=e^x+x^e. Its derivative is e^x+e x^(e−1). Dividing this derivative by e gives e^(x−1)+x^(e−1), exactly the numerator. Thus the integrand is g′/(eg), and integration gives $(1/e)\ln|e^x+x^e|+C$, matching C on the domain of the stated real powers.

GATE 2023 MN – Question 39

Explanation

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