# How to solve this double integral? $\int_0^{\pi/2}\int_{x}^{\pi/2} \frac{cos{y}} y\, dy ~dx$

$$\int_0^{\pi/2}\int_{x}^{\pi/2} \frac{cos{y}} y\, dy ~dx$$

I'm not sure how to go about integrating this. I think that I need to integrate with respect to y first because the lower bound is x, which would be used for the integration with respect to x.

But, I don't know how to go about integrating the function with respect to y. How can this be done?

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Change the limits, and integrate w.r.t. $x$ first. –  i707107 Apr 2 '13 at 8:33
Won't that just give me a constant multiplied by the function itself to integrate? –  bubblyicecream Apr 2 '13 at 8:50
It will change your bounds and not be 'just a constant.' –  mixedmath Apr 2 '13 at 8:52
Possible duplicate: Help with some double integral –  M. Strochyk Apr 2 '13 at 9:09

$$\int\limits_0^{\pi/2}\int\limits_x^{\pi/2}\frac{\cos y}{y}dydx=\int\limits_0^{\pi/2}\int\limits_0^y\frac{\cos y}{y}dxdy=\left.\int\limits_0^{\pi/2}\cos y\,dy=\sin x\right|_0^{\pi/2}=1$$
Hint: define $f(x):=\int_x^{\frac\pi 2}\frac{\cos y}ydy$, and integrate by parts: $$\int_0^{\frac\pi 2}f(x)dx=\frac{\pi}2f\left(\frac{\pi}2\right)-\int_0^{\frac{\pi}2}xf'(x)dx.$$