I want to calculate ∫ C + ( 2 , 2 ) e z ( z − 2 ) 2 d z and ∫ 0 ∞ 1 1 + x x d x

Question

I want to calculate      ∫                  C        +            (      2      ,      2      )                  e              z                    (      z      −      2              )        2              d  z        and          ∫          0              ∞            1          1      +      x              x              d  x

lelestalis80d · Accepted Answer

The second integral may be transformed into a standard integral used in residue theory by the substitution   u  =      x          3              /            2       to get      ∫    0          ∞                  d      x              1      +      x              x              =      2    3        ∫    0          ∞        d  u              u              −        1                  /                3                    1      +      u      The integral on the RHS may be evaluated by considering      ∮    C    d  z              z              −        1                  /                3                    1      +      z      The result is  (  1  −      e          −      i      2      π              /            3        )      ∫    0          ∞        d  u              u              −        1                  /                3                    1      +      u        =  i  2  π      e          −      i      π              /            3      I leave it to the reader to work the algebra. The result is      ∫    0          ∞        d  u              u              −        1                  /                3                    1      +      u        =            2      π              3      and therefore      ∫    0          ∞                  d      x              1      +      x              x              =            4      π              3              3

I want to calculate int_(C^(+)(2,2)) e^(sqrt(z))/(z-2)^2dz and int_0^infty 1/(1+x sqrt(x))dx

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