Efficient Integration of x^3 / (x+1)^10: Tips & Tricks"

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1. Integrate x^3 / (x+1)^10



Tried substitution with u = x+1, or x+1 cubed, but no go
 
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You could just write it as x^3(x+1)^{-10} and integrate by parts a few times.
 
O.J. said:
Tried substitution with u = x+1, or x+1 cubed, but no go

If u = x+1, then x = u-1, isn't it?
 

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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