Integrate 1 / (x+1)^2 square root (x^2+2x+2)

Answers (1)

Solution:  

         I=int frac1(x+1)^2sqrtx^2+2x+2dx

                x^2+2x+2=(x+1)^2+1

  Let  us  put     x+1=t;   then 

            I=int frac1t^2sqrtt^2+1dt

Again we have integral  of the form  I.  make the substitution   t=sinh z.

            dt =cosh z dz hspace1cmsqrtt^2+1=sqrt1+sinh^2z=cosh z

Hence ,       I=fraccosh z dzsinh^2zcosh z=int frac1sinh^2zdz=-coth z+c

                  I=-fracsqrt1+t^2t+c=-fracsqrtx^2+2x+2x+1+c.

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