Changing Limits of Integration affects variable substitution?

DrMath
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Homework Statement



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Homework Equations





The Attempt at a Solution

 

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not exactly sure what the question is...

couple of points though, it is usually easier to use a different dummy variable when you use a subtitution

say you have
\int_b^a dt f(t)

and want to change to u(t) = t/2, then
t=2u
dt = 2.du
u(a) = a/2
u(b) = b/2

so the integral becomes
\int_{b/2}^{a/2} 2.du f(2u)

see how this compares with your case...
 
thanks
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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