Converting single integral to double integral

[Amad27]

Homework Statement

Please refer to : http://math.stackexchange.com/quest...x-arctan-frac1x-mathrm-dx-fra/1069065#1069065

The answer by @venus.

What is the procedure in converting that single integral, dividing it into parts, and making it a double integral?

And also, how Venus took $\sin(x)$ and brought it inside the first integral, and interchanging the integrals?

What is the criterion?

I am very interested in this.

Any **links** advice or comment is very helpful.

Thanks!

Homework Equations

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The Attempt at a Solution

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I am looking for a general method, that's all.

 

[BiGyElLoWhAt]

you're integrating with respect to y first, and therefore sin(x) is a constant with respect to your first integral, as is the integral of sin(x). You can pull it out of the second integral or put it into the integral.

The second integral came from the fact that ##\int \frac{x}{x^2 +y^2} = arctan(\frac{1}{x})## and that's in the original integral. Just another way of expressing the same thing.

If I had ##\int xy dx##, where y was a function of x let's say equal to x, we could write that as ##\int x \int (\frac{d}{dx})(y)## since the integral cancels the derivative acting on y. This then becomes ##\int \int xdydx## and the first integral you calculate is dy, (x is a constant with respect to y)so you end up back at... well, I'll let you take it from here. On math.stack, y = arctan(\frac{1}x{x}