Evaluate: integral from 0 to (pi)/(2) of integral from 0 to x of x * sin(y) y x

Expression: $\int_{ 0 }^{ \frac{ π }{ 2 } } \int_{ 0 }^{ x } x \times \sin\left({y}\right) \mathrm{d} y \mathrm{d} x$

To make the calculation easier, change the order of integration

$\int_{ 0 }^{ \frac{ π }{ 2 } } \int_{ y }^{ \frac{ π }{ 2 } } x \times \sin\left({y}\right) \mathrm{d} x \mathrm{d} y$

To evaluate the iterated integral, first evaluate the inner integral

$\int_{ 0 }^{ \frac{ π }{ 2 } } \frac{ {π}^{2} \times \sin\left({y}\right) }{ 8 }-\frac{ \sin\left({y}\right) \times {y}^{2} }{ 2 } \mathrm{d} y$

Evaluate the definite integral

\begin{align*}&-\frac{ π }{ 2 }+\frac{ {π}^{2} }{ 8 }+1 \\&\approx0.662904\end{align*}

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