definite integral + change of variable

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Opus89

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Sep 21, 2008
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I have to integrate the definite integral of (x + 1)/(x^2 + 1) dx. The upper bound is 2, the lower bound is 0. I split the integral up in two parts- x/(x^2 + 1) + 1/(x^2 + 1).
I was able to solve the first part and I got (1/2)ln5. I need some help getting started with the second part, 1/(x^2 + 1). I tried to let u = x^2 + 1. Thus, du = 2xdx. I tried to solve for dx and I got du/2x = dx. I don't see how this can work. I can't use xdx = du/2 because i don't have an x in the numerator. Can someone give me some advice please? Thanks
 
Notice what it is without working it out.

HINT: What is the derivative of arctan?.
 
The derivative of arctan u = u'/(1 + u^2). I sort of see the connection but the u' is throwing me off. If the u' was just u it would look exactly like 1/(x^2 + 1).
 
\(\displaystyle \frac {d[\tan^{-1}x]}{dx} \, = \, \frac {1}{1+x^2}\)
 
So the integral of 1/(x^2+1) = archtan? So I evaluate archtan at 2 and 0?
 
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