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June 19th, 2014, 01:41 PM   #1
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Changing the intergrator

Hi there,

I am studying for an exam, but I'm having problems with an integral, which is:

1/2 * integral(0 to pi)(y sin y^2) dy = 1/4 * integral(0 to pi)(sin y^2) dy^2

I have had some material on Riemann-Stieltjes integrals, which I can apply here, but I should not be using that for this course, as it is a more introductory course. I am curious if there are simple rules to change the integrator.

Thanks in advance.
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June 19th, 2014, 04:09 PM   #2
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$\newcommand{\d}[1]{\mathbb{d}{#1}}$
Write $x = y^2$ then $\d{x} = 2y \d{y}$, $y = 0 \implies x=0$ and $y= \pi \implies x=\pi^2$
And wee thus have$$2 \int \limits_{0}^{\pi}y \sin{y^2} \d{y} = \int \limits_{0}^{\pi^2} \sin{x} \d{x}$$
which has an odd upper limit, but otherwise all is well.
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