GATE 2025 ST – Question 25

Calculus · Multiple Integrals · 1 mark · Numerical answer

Evaluate $I=\pi^2\int_0^1\int_0^1 y^2\cos[\pi(1+xy)]dx\,dy$ (integer).

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Correct answer: -1 to -1

Explanation

Integrate in x: for y>0 the inner integral is y[sinπ(1+y)−sinπ]/π=−y sinπy/π. Thus I=−π∫₀¹y sinπy dy. Integration by parts gives ∫₀¹y sinπy dy=1/π. Therefore **I=−1**.

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