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How does one compute the value the following integral $$ \int_0^1 \int_y^1 \frac{\sin x}{x} dx dy$$

Direct integration involves a non-elementary function (erfc), so a change of variables is necessary. However, I can't figure out any useful one. Any suggestions?

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Try switching the limits of integration. –  Eric Stucky Oct 21 '12 at 6:37
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2 Answers

$$ \begin{align*} \int_{0}^{1}\int_{y}^{1}\frac{\sin x}{x}\,dxdy &=\left[ y \int_{y}^{1}\frac{\sin x}{x}\,dx \right]_{0}^{1} - \int_{0}^{1} y\frac{d}{dy}\int_{y}^{1}\frac{\sin x}{x}\,dxdy\\ &=\int_{0}^{1}\sin y \, dy. \end{align*} $$

Now the rest is clear. It is not obvious from the calculation, but in general integration by parts can be thought as a special case of the interchange of the order of integration, the method which Eric Stucky pointed out.

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$$\int_0^1 \int_y^1 f(x,y) dx dy=\int_0^1 \int_0^x f(x,y) dy dx $$

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