Solving ∫(x^7/(1+x^16)dx: Tips & Tricks

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Homework Statement


∫(x^7/(1+x^16)dx

Homework Equations





The Attempt at a Solution


I feel like it's so simple. I have tried making u equal to 1+x^16 and then trying to find some sort of chain I can use to make the equation work but no luck. When i saw that arctan was in the correct answer I was lost. Any help appreciated.
 
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Never mind. I just had to look at it for a second more.
 
And you let u= x^8?
 
Yes.
 

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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