GATE 2018 CE (CE1) – Question 36

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 2 marks · Multiple choice

Evaluate $\int_0^\pi x\cos^2x\,dx$.

$\pi^2/8$
$\pi^2/4$
$\pi^2/2$
$\pi^2$

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Correct answer: (B) $\pi^2/4$

Explanation

Use $\cos^2x=(1+\cos2x)/2$. The first term integrates to $\pi^2/4$. Integration by parts gives zero for $\int_0^\pi x\cos2x\,dx$ because endpoint sine terms vanish and cosine terms cancel. Answer B.