advanced integration problem ∫x²/(√25 - x²)

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You need to solve the integral:`int (x^2 dx)/sqrt(25 - x^2)`

You need to write the integral such that:

`int x*(x/sqrt(25 - x^2))*dx`

You may use integration by parts such that:

`u = x=gt du = dx`

`` `dv = xdx/sqrt(25 - x^2) =gt v = -sqrt(25 - x^2)/2`

`int x*(x/sqrt(25 - x^2))*dx = -(xsqrt(25 - x^2))/2 + int sqrt(25 - x^2)dx/2`

`int x*(x/sqrt(25 - x^2))*dx = -(xsqrt(25 - x^2))/2 + (1/4)*(xsqrt(25 - x^2) + 25arcsin (x/5)) + c`

Evaluating the integral yields: `int (x^2 dx)/sqrt(25 - x^2) =-(xsqrt(25 - x^2))/2 + (1/4)*(xsqrt(25 - x^2) + 25arcsin (x/5)) + c`

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