integrate cos-1((1-x2) / (1+x2)) dx using by parts

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Please find below the solution to the asked query:

We have:I=cos-11-x21+x2.dxWe know that:cos-11-x21+x2=2tan-1xI=2tan-1x.dx=21.tan-1x.dxUsing integration by part taking 1 as second function we get:I=2tan-1x1.dx-ddxtan-1x1.dxdx=2tan-1x.x-11+x2.xdx=2x.tan-1x-x1+x2dx=2x.tan-1x-2x1+x2dxNow in 2x1+x2dx  if we take 1+x2=t, then 2x.dx=dt, hence, we getdtt=lnt=ln1+x2I=2x.tan-1x-ln1+x2+C, where C is integration constant. cos-11-x21+x2.dx=2x.tan-1x-ln1+x2+C

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