Evaluating definite integrals for the area of the regoin

1. The problem statement, all variables and given/known data

Evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.



[itex]f(x) = \frac{1}{x^2+1}[/itex] at the point (1,1/2)





2. Relevant equations







3. The attempt at a solution



So far I have obtained that the tangent line is [itex]y = 1- \frac{1x}{2}[/itex]. I'm having difficultly integrating for the area.



I've worked out that [itex]A = \int_1^0 [\frac{1}{x^2+1} - (\frac{-1x}{2} +1)]\,dx[/itex].

I'm not really sure what to do from here. I know that i'm suppose to fit the first fraction into the integral of arctan, however I can not get the top to be equal to du. In addition, I thought I should use the power rule for the second part of the fraction to obtain x^2/2 but the answer in my book states that it's x^2/4 which is confusing me. Any help would be appreciated, but please let me know what rules you are using to obtain your solution so it's easier for me to understand. Thanks.