INTEGRATE

x²+1/ (x+1)²

Hi Abhishek,

MAke the sub x+1 = t.

So the integral will be {t^2 - 2t + 2}/t^2 = 1 - 2/t + 2/t^2

Which is directly integrable.

Best Regards,

Ashwin (IIT Madras).

integral of   X2+1/(x+1)2

 ans..    x-2log(x+1)-2/x+1  +C

ANS IS X-2[(1/X+1)+LN(X+1)]

∫{(x²+1)/(x+1)²}dx

=∫(x²+1+2x-2x)/(x²+1+2x)}dx

=∫{(x²+1+2x)/(x²+1+2x)}dx-∫{(2x+2-2)/(x²+1+2x)}dx

=∫1.dx-∫{(2x+2)/(x²+1+2x)}dx+∫{2/(x+1)²}dx

=x-∫(1/t)dt+2(-1)(x+1)^-1 ...................................[Let x²+1+2x=t→(2x+2)dx=dt]

=x- ln ιtι - 2/(x+1) + c

=x- ln ιx²+1+2xι - 2/(x+1) + c...............where ιxι represents logarithm of x.

write x2+1 as (x+1)2-2x

integrate