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Question

xx2+2x+2 x+1 dx

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Solution

We have,I= x dxx2+2x+2 x+1= x dxx+12+1 x+1Putting x+1=t2x=t2-1Diff both sidesdx = 2t dtI= t2-12t dtt22+1 t=2 t2-1dtt4+1Dividing numerator and denominator by t2I=21-1t2t2+1t2dt

=21-1t2dtt2+1t2+2-2=2 1-1t2dtt+1t2-22Putting t+1t=p1-1t2dt=dpI=2 dpp2-22=2×122log p-2p+2+C=12log p-2P+2+C=12log t+1t-2t+1t+2+C=12log t2-2t+1t2+2t+1+C=12log x+1-2x+1+1x+1+2x+1+1+C=12log x+2-2x+1x+2+2x+1+C

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