Weird question, antiderivative and integralGiven that F(x) is the antiderivative to the function f ( x ) = x 2 ∗ l n ( x ) which satisfies that F ( 1 ) = 7 / 9, calculate F(2).How does one do that? I first figured that I would calculate the general F(x), use the condition that F ( 1 ) = 7 / 9 to find C, then calculate F(2)... but it seemed too messy, and I am supposed to do this quickly and without calculators. Is there a trick or something I am missing?

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Weird question, antiderivative and integralGiven that F(x) is the antiderivative to the function   f  (  x  )  =      x    2    ∗  l  n  (  x  ) which satisfies that   F  (  1  )  =  7      /    9, calculate F(2).How does one do that? I first figured that I would calculate the general F(x), use the condition that   F  (  1  )  =  7      /    9 to find C, then calculate F(2)... but it seemed too messy, and I am supposed to do this quickly and without calculators. Is there a trick or something I am missing?

Ellie Meyers · Accepted Answer

Step 1You can find F by using partial integration:                    ∫                  x          2                ⋅        ln        ⁡        x        d        x        =                              x            3                    3                ln        ⁡        x        −        ∫                              x            3                    3                ⋅                  1          x                d        x        =                              x            3                    9                (        3        ln        ⁡        x        −        1        )        +        C        .            .Step 2Hence       7    9    =  F  (  1  )  =  −      1    9    +  C  ,, so   C  =      8    9  . Now it is easy to see that   F  (  2  )  =      8    9    (  3  ln  ⁡  2  −  1  )  +      8    9    =      8    3    ln  ⁡  2..

Carley Haley · Accepted Answer

Step 1Your approach is sound.Try integration by parts:   u  =  ln  ⁡  x    ⟹    d  u  =      1    x      d  x  .    d  v  =      x    2      ⟹    v  =      1    3        x    3    ∫      x    2    ln  ⁡  x    d  x  =      1    3        x    3    ln  ⁡  x  −      1    3    ∫      x    2      d  xStep 2The remaining integral is a cinch. Note that F(1) evaluates nicely (since   ln  ⁡  (  1  )  =  0), which you can use to solve the invariable constant of integration.I don&#039;t think there&#039;s any way around actually integrating and evaluating.

Weird question, antiderivative and integral Given that F(x) is the antiderivative to the function

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