Evaluate dx/2x-x2, where c is an integration constant.

GATE 2022 BM – Question 12

Engineering Mathematics · Calculus · 1 mark · Multiple choice

Evaluate $\int dx/\sqrt{2x-x^2}$, where c is an integration constant.

$\sin^{-1}(x-1)+c$
$\cos^{-1}(x-1)+c$
$\sin^{-1}(x/2)+c$
$\cos^{-1}(x/2)+c$

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Show answer and explanation

Correct answer: (A) $\sin^{-1}(x-1)+c$

Explanation

Complete the square: 2x−x²=1−(x−1)². Substitution u=x−1 reduces the integral to ∫du/√(1−u²)=arcsin u+c. Thus A. The derivative of arccos has a minus sign, so B does not match.

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