Quick integrating question (part of a series question)

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Quick integrating question...(part of a series question)

Still studying for my exam...


Came across this example and I don't know what the method is for integrating this type of problem? (I'm using it to do the integral test for series). How did they get rid of the x^3? thanks for the help :-)

int.jpg
 
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Well what's the derivative of -1/3 * exp(-x^3)?
 
Or: let u= x2 so that du= 3x2 dx.
 
ahhh...i see it now. 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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