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Question

tan-1 x dx

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Solution

Let I= tan-1 x dx = x.tan-1 x dxxLet x=t12xdx=dt=dxx=2dtI=2 tII.tan-1 tI dt =2 tan-1tt dt-ddttan-1tt dtdt =2 tan-1 t.t22- 11+t2.t22dt =tan-1 t.t2- t21+t2 dt =tan-1 t.t2- 1+t2-11+t2dt =tan-1 t.t2- dt+dt1+t2 =tan-1 t.t2-t+tan-1 t+C x=t =tan-1 x.x-x+tan-1 x+C =x+1 tan-1 x-x+C

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