Evaluating ∫ ( 3 5 − 8 x ) d xI am unsure how to get the antiderivative of f ( x ) = 3 5 − 8 x I know the answer is − 8 ln ⁡ ( | x | ) + C + 3 x 5 as it says in my textbook.But I am unsure how I get something like ln(x) in the answer? Could someone guide me through this? I am totally lost as for other antiderivatives I have no problem but this stuns me.

Question

Evaluating   ∫      (          3      5        −          8      x        )       d  xI am unsure how to get the antiderivative of   f  (  x  )  =      3    5    −      8    x  I know the answer is   −  8  ln  ⁡  (      |    x      |    )  +  C  +            3      x        5   as it says in my textbook.But I am unsure how I get something like ln(x) in the answer? Could someone guide me through this? I am totally lost as for other antiderivatives I have no problem but this stuns me.

Brianna Sims · Accepted Answer

Explanation:The antiderivative of             1      x       is   ln  ⁡      |    x      |  . You can break up the antiderivatives by addition, subtraction, or you can pull scalar multiples out to the front. Since   −            8      x       is just   −  8  ⋅            1      x      , the antiderivative is   −  8  ln  ⁡      |    x      |    +  C.

Lughettitbn · Accepted Answer

Step 1You need to use the fact the derivative is linear            (              k            ⋅              f        (        x        )            +              j            ⋅              g        (        x        )            )        ′    =      k    ⋅            f      ′        (    x    )    +      j    ⋅            g      ′        (    x    )  You seem to know that   (  x      )    ′    =  1  (  log  ⁡  x      )    ′    =            1      x      Step 2Then, using the above, can you find a primitive of   f  (  x  )  =      3    5    ⋅      1    +  (  −  8  )  ⋅            1      x         ?

Evaluating ∫ <mrow> ( <mfrac> 3 5 </mfrac> &#x2212

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New Questions in Calculus 1