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The value of the integral

∫ 1 { x2 + x2013/(x2 + 2IxI +1)}dx
(1) positive
(2) negative
(3) zero
(4) infinite

dx is
the question is not clear... x2013 ?? what does it mean ? what is the expression in numerator... x square + x2013 or only x2013 ? use parentheses if required to clearly write the expression
it is x to the power 2013 and x square .. & limit of the integration is from 1 to -1 .. please help me



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I suppose the given expression inside the integral is as below.

I= \int\limits^{1}_{x=-1} [{x^2+\frac{x^{2013}}{x^2+2|x| +1}}] \, dx \\\\I= \int\limits^{1}_{-1} {x^2} \, dx + \int\limits^{1}_{-1} {x*\frac{(x^2)^{1006}}{x^2+2|x| +1}}} \, dx \\\\=\frac{1}{3}[x^3]_{-1}^{1}+\int\limits^{0}_{-1} {x*\frac{(x^2)^{1006}}{x^2+2|x| +1}}} \, dx +\int\limits^{1}_{0} {x*\frac{(x^2)^{1006}}{x^2+2|x| +1}}} \, dx \\\\I=\frac{2}{3}-\int\limits^{-1}_{0} {x*\frac{(x^2)^{1006}}{x^2+2|x| +1}}} \, dx +\int\limits^{1}_{0} {x*\frac{(x^2)^{1006}}{x^2+2|x| +1}}} \, dx \\\\I=2/3

Let f(x) be function inside ONE Integral on the RHS.  Then let X= -x.

F(x)=x*\frac{(x^2)^{1006}}{x^2+2|x| +1}\\\\I=2/3 - \int\limits^{1}_{X=0} {F(X)} \, dX + \int\limits^{1}_{x=0} {F(x)} \, dx\\\\Hence,\ I=\frac{2}{3}

In the given integral G(x) = \frac{x^{2013}}{x^2 + 2 |x| + 1} is anti symmetric function.  so its image wrt y axis.  So  G(x) = - G(-x).  Hence,  the area under the curve G(x) for x > 0 is equal and opposite to the area under the curve for  x < 0. 

1 5 1
click on thanks button above please
what is antisymmetric function?
how do u know that G(X) = -G(-X)
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