Use partial fractions to solve the integralint frac{x^2-x+20}{x^3+5x}dx

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grocbyntza · Accepted Answer

∫                    x        2            −      x      +      20                      x        3            +      5      x        d  x    ∫                    x        2            −      x      +      20              x      (              x        2            +      5      )        d  x        A    x    +            B      x      +      C                      x        2            +      5        =                    x        2            −      x      +      20              x      (              x        2            +      5      )          A  (      x    2    +  5  )  +  C  x  +  5  A  =      x    2    −  x  +  20    A  +  B  =  1  ,  C  =  −  1  ,  5  A  =  20    B  =  −  3  ,  A  =  4    I  =  ∫  (      4    x    +            −      3      x      −      1                      x        2            +      5        )  d  x    =  4  ln  ⁡  x  −      3    2    ∫            2      x      d      x                      x        2            +      5        −  ∫            d      x                      x        2            +      5          =  4  ln  ⁡  x  −      3    2    ln  ⁡  (      x    2    +  5  )  −      1          5            tan    ′    ⁡  (      x          5        )  +  C  [  in the 2nd term let       x    2    +  5  =  v           x  d  x  =  d  x      /    2  ]    ∫            d      x        x    =  ln  ⁡  x  +  C    ∫            d      x                      x        2            +              a        2              =      1    a        tan          −      1        ⁡  (      x    a    )    I  =  4  ln  ⁡  x  −      3    2    ln  ⁡  (      x    2    +  5  )  −      1          5            tan          −      1        ⁡  (      x          5        )  +  C

Use partial fractions to solve the integral int (x^2-x+20)/(x^3+5x)dx

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