I am trying to calculate the integral ∫ − ∞ ∞ a + x b 2 + ( a + x ) 2 1 1 + c ( a − x ) 2 d xwhere { a , b , c } ∈ R . I have looked in a table of integrals for rational functions, but with no luck. Is there a smart trick I can utilize?

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I am trying to calculate the integral      ∫          −      ∞              ∞                          a        +        x                              b          2                +        (        a        +        x                  )          2                            1              1        +        c        (        a        −        x                  )          2                      d  xwhere   {  a  ,  b  ,  c  }  ∈      R  . I have looked in a table of integrals for rational functions, but with no luck. Is there a smart trick I can utilize?

podesect · Accepted Answer

Hint: Use partial fraction to write the integral as                    a        +        x                              b          2                +        (        a        +        x                  )          2                            1              1        +        c        (        a        −        x                  )          2                      =            A      +      B      x                      b        2            +      (      a      +      x              )        2              +            C      +      D      x              1      +      c      (      a      −      x              )        2              ,then you can write it as      A                  b        2            +      (      a      +      x              )        2              +      C          1      +      c      (      a      −      x              )        2              +            B      x                      b        2            +      (      a      +      x              )        2              +            D      x              1      +      c      (      a      −      x              )        2              ,and finally make the change of variables   x  −  a  =  t and   x  +  a  =  t and things will be easy.

Kassandra Ross · Accepted Answer

Potential Hint:      ∫          −      ∞              ∞                  a              (                  b          2                +        (        a        +        x                  )          2                )        (        1        +        c        (        a        −        x                  )          2                )              −            x              (                  b          2                +        (        a        +        x                  )          2                )        (        1        +        c        (        a        −        x                  )          2                )              d  xThe first part can be solved using a chain rule like method. The second half can be solved using integration by parts with   u  =  x i.e.   u  ′  =  1Hope this helps!

I am trying to calculate the integral <msubsup> ∫ <mrow class="MJX-T

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