Antiderivative of a polynomial fractionI am trying to solve this problem and I can't seem to understand where I am going wrong. My understanding of antiderivatives is this formula: ∫ x n d x = x n + 1 n + 1 + cThe problem I am trying to solve is: f ( x ) = x 5 − x 3 + 8 x x 4 I think there are two possible parts that I am getting wrong, I started the problem by separating them into individual fractions and solving each fraction at a time. I think I get it wrong at either the ln(x) or 8 x − 2 − 2 . Here is my attempt: f ( x ) = x 5 x 4 − x 3 x 4 + 8 x x 4 = x − 1 x + 8 x 3 Then from here I get the antiderivative: F ( x ) = x 2 2 − ln ⁡ ( x ) + 8 x − 2 − 2 + c = x 2 2 − ln ⁡ ( x ) − 4 x 2 + cI keep getting the wrong answer when I plug it in, am I missing some fundamental piece of information about antiderivatives which is causing me to mess up the ln(x) or is my algebra wrong and I mess up around 8 x − 2 − 2 ?

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Antiderivative of a polynomial fractionI am trying to solve this problem and I can&#039;t seem to understand where I am going wrong. My understanding of antiderivatives is this formula:  ∫      x          n            d    x  =            x              n        +        1                    n      +      1        +  cThe problem I am trying to solve is:  f  (  x  )  =                    x        5            −              x        3            +      8      x              x      4      I think there are two possible parts that I am getting wrong, I started the problem by separating them into individual fractions and solving each fraction at a time. I think I get it wrong at either the ln(x) or             8              x                  −          2                            −      2      . Here is my attempt:   f  (  x  )  =            x      5              x      4        −            x      3              x      4        +            8      x              x      4        =  x  −      1    x    +      8          x      3      Then from here I get the antiderivative:  F  (  x  )  =            x      2        2    −  ln  ⁡  (  x  )  +            8              x                  −          2                            −      2        +  c  =            x      2        2    −  ln  ⁡  (  x  )  −      4          x      2        +  cI keep getting the wrong answer when I plug it in, am I missing some fundamental piece of information about antiderivatives which is causing me to mess up the ln(x) or is my algebra wrong and I mess up around             8              x                  −          2                            −      2      ?

postojahob · Accepted Answer

Explanation:The domain of the natural logarithm is   (  0  ,  ∞  ). However, 1/x is defined on   (  −  ∞  ,  0  )  ∪  (  0  ,  ∞  ), so should be the derivative of some function defined on this larger set. That &quot;some function&quot; is ln|x|. That is             d                      d            x        ln  ⁡      |    x      |    =      1    x    ,    x  ≠  0 so   ln  ⁡      |    x      |    +  C  =  ∫      1    x          d    x  ,    x  ≠  0  .

cooloicons62 · Accepted Answer

Explanation:The problem is in the integration of x:  ∫  x  −      1    x    +      8          x      3          d  x  =            x      2        2    −  ln  ∣  x  ∣  −      4          x      2        +  C

Antiderivative of a polynomial fraction I am trying to solve this problem and I can't seem to under

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New Questions in Calculus 1