The value of the integral
∫ 1 { x2 + x2013/(x2 + 2IxI +1)}dx
-1
(1) positive
(2) negative
(3) zero
(4) infinite




dx is

1
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

Answers

2015-06-18T23:28:10+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
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)