Antiderivative of d y d x Find the antiderivative of d y d x = e 2 x − x and y = 5 when x = 0.the antiderivative formula is e k x d x = 1 k e k x + c.I did it, d y d x = e 2 x − x and y = 5 when x = 0. d y d x = 1 2 e 2 x + c but the right answer is y = 1 2 ( e 2 x − x 2 + 9 )

Question

Antiderivative of             d      y              d      x      Find the antiderivative of             d      y              d      x       =       e          2      x        −  x and   y  =  5 when   x  =  0.the antiderivative formula is       e          k      x        d  x =       1    k        e          k      x        +  c.I did it,             d      y              d      x       =       e          2      x        −  x and   y  =  5 when   x  =  0.            d      y              d      x        =      1    2        e          2      x        +  c but the right answer is   y  =      1    2    (      e          2      x        −      x    2    +  9  )

Rebekah Zimmerman · Accepted Answer

Step 1You made two major errors. First, you found the antiderivative only of the exponential term and forgot all about the   −  x term: you should have got   y  =      1    2        e          2      x        −            x      2        2    +  c    ..Note that this is y, not dy/dx.Step 2Secondly, you found all of the antiderivatives, but the problem requires you to find the specific antiderivative such that   y  =  5 when   x  =  0. Take the corrected general antiderivative,  y  =      1    2        e          2      x        −            x      2        2    +  c    ,,and substitute   y  =  5 and   x  =  0 to get  5  =      1    2        e    0    −      0    2    +  c  =      1    2    +  c    ;;clearly we must have   c  =  5  −      1    2    =      9    2      ,, and the unique antiderivative that satisfies the extra condition is   y  =      1    2        e          2      x        −            x      2        2    +      9    2    =      1    2        (          e              2        x              −          x      2        +    9    )      .

Antiderivative of d y </mrow> d x </mrow> </mfrac>

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