Solving Tricky Integral Homework Problem

[Appa]

Homework Statement

I should find this integral:
\int^b x(1/\Pi)(1/(1+x²)dx
^-b

Homework Equations

\int1/(1+x²)dx = arctan(x)

The Attempt at a Solution

The Only thing I've succeeded in doing is to take the 1/\Pi and put it in front of the integral like this:
(1/\Pi)\int^b (x/(1+x²)dx
^-b
And I know that the integral of 1/(1+x²) equals arctan(x) but how could that help me? I've tried to use the equation
\intf(x)g(x)dx = F(x)g(x) - \intF(x)g'(x)dx
but I can't compute the integral of arctan(x).
Could someone help me?

 

[Mark44]

Is this the integral?
\frac{1}{\pi}\int_{-b}^b{\frac{x}{x^2 + 1}dx

If so, you can use a simple substitution, u = x^2 + 1, and du = 2xdx

 

[Mark44]

BTW, you should have posted this in the Calculus & Beyond forum. This problem clearly falls in that area.

 

[NoMoreExams]

Seems kind of weird to be doing that integral from -b to b, whether or not b > 0 or < 0 you get into complex numbers.

 

[Gib Z]

Perhaps the easiest way is to observe this function is odd, and look at the interval of integration.

 

[HallsofIvy]

Seems kind of weird to be doing that integral from -b to b, whether or not b > 0 or < 0 you get into complex numbers.

No, you don't. You are squaring not taking square roots. As Mark44 said, use the substitution u= x²+ 1. Or, even simpler, use Gib Z's suggestion. This is really a very simple integral.

 

[NoMoreExams]

Oh duh, good point.

 

[Appa]

Yeah, sorry, I got it myself pretty soon after posting this. It feels like the more I study maths, the more I forget..!