- #1

Saitama

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## Homework Statement

Consider ##f(x)=4x^4-24x^3+31x^2+6x-8## be a polynomial function and ##\alpha, \beta, \gamma, \delta## are the roots of the equation ##f(x)=0##, where ##\alpha < \beta < \gamma < \delta##. Let sum of two roots of the equation f(x)=0 vanishes. Then the value of

[tex]\int_{2\alpha}^{2\beta} \frac{x^{\delta+1}-5x^{\gamma+1}+2\beta |x|+1}{x^2+4\beta |x|+1}dx[/tex]

## Homework Equations

## The Attempt at a Solution

I succeeded in finding the roots. ##\alpha=\frac{-1}{2}, \beta=\frac{1}{2}, \gamma=2## and ##\delta=4##.

Therefore, the integral is

[tex]\int_{-1}^{1} \frac{x^5-5x^3+|x|+1}{x^2+2|x|+1}dx[/tex]

I don't know how to proceed further from here. The only thing I can think of is to write the integral in two parts,

[tex]\int_{-1}^{0} \frac{x^5-5x^3-x+1}{(x-1)^2}dx +\int_{0}^{1} \frac{x^5-5x^3+x+1}{(x+1)^2}dx [/tex]

Any help is appreciated. Thanks!