Discussion
The value of the integral \(I = \int_{0}^{1} \int_{0}^{1} x \cos(xy)\, dx\, dy\) is
Answer: A
The integral equals \(1 - \cos(1)\).
The limits are constants, so the order of integration can be swapped. Integrating with respect to \(y\) first is easy, because \(x\) is exactly the factor produced by differentiating \(\sin(xy)\) with respect to \(y\):
\[ \begin{aligned} \int_{0}^{1} x\cos(xy)\, dy &= \Big[ \sin(xy) \Big]_{y=0}^{y=1} = \sin x \\ I &= \int_{0}^{1} \sin x\, dx \\ &= \Big[ -\cos x \Big]_{0}^{1} = 1 - \cos 1 \end{aligned} \]
Numerically this is about 0.46. \(1 + \cos 1\) has the sign of the cosine term wrong, and the two options with \(\sin 1\) come from integrating \(\cos\) instead of \(\sin\) in the last step. Integrating in the written order (first in \(x\)) needs integration by parts and gives the same result with more work.
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