Subtopic: Maxima and minima of function with several independent variables
Question Statement:
The value of the integral $I = \int_{0}^{1} \int_{0}^{1} x \cos(xy) \, dx \, dy$ is:
Options:
(A) 1 − cos
(B) 1 + cos
(C) 1 − sin
(D) 1 + sin
Official Answer Key:
A
Step-by-Step Solution & Explanation
Integrating with respect to x first is messy (requires integration by parts). Reverse the order and integrate with respect to y first, treating x as a constant:
∫₀¹ x cos(xy) dy = x · [sin(xy)/x]₀¹ = [sin(xy)]₀¹ = sin(x) − 0 = sin x
Then:
I = ∫₀¹ sin x dx = [−cos x]₀¹ = −cos(1) + cos(0) = 1 − cos(1)
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