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## Homework Statement

Integral (pi

^{1/2}, 0) of Integral (pi

^{1/2}, y)

**sin(y**

^{2}) dxdy## Homework Equations

This one is interesting because it can't be integrated as is (at least not at the level of my course) but I think with some rearranging it can be done. I was wondering if anyone could verify my method.

BTW, I'm kinda new, so I don't know exactly how to make the little integral symbol. If that's too hard to read, it's a double integral defined on the inside between "y" and the square root of pi, and defined on the outside between 0 and the square root of pi. The function is sin(y

^{2}), with dx, then dy. That is, I believe 0 < y < pi

^{1/2}and y < x < pi

^{1/2}.

## The Attempt at a Solution

Alright so as is, it's not an elementary function and cannot be worked with, I believe. I rearranged the "parameters" to define x as 0 < x < pi

^{1/2}and 0 < y < x. Then, "dx" and "dy" switch positions in the original equation, and I integrated sin(y

^{2}) with respect to "x" first. Simply put, I got xsin(y

^{2}). Then, integrating with respect to "y" by parts, I got xsin(y

^{2}) - [(1/2)(x

^{2}) * -(2y)(cos(y

^{2}))]. Without doing all the math here, I'll tell you I ended up with pi

^{2}.

I feel I got the problem right all the way until the final integration. I'm mostly afraid I messed up the final step in integration. If anyone would like to look at my process, that would be great.

PS -- if anyone knows how to directly integrate a non-elementary function, that would be cool to see.