Question

Let F(x) be an antiderivative of `(2(lnx)^4)/(3x)`, If F(2)=0, then F(8)=?

The correct answer is 5.163 but I don't understand how to get it.
Do I have to anti-differentiate this function and then plug '8' in?

Answer 1

Yes, you're correct. See below.

Explanation:

We are given:

`f(x) = (2(ln(x))^4)/(3x)`

We can integrate. Let `tau equiv ln(x)`.

`F(x) = int" " (2tau^4)/(3) d tau`

`F(x) = 2/15 tau^5 +C`

`F(x) = 2/15 (ln(x))^5 + C`

where `C` is some arbitrary integration constant.

We are given more information, that `F(2) = 0`:

`F(2) = 2/15 (ln(2))^5 + C = 0`

`C = -2/15 (ln(2))^5`

Hence, we find:

`F(x) = 2/15(ln(x))^5 - 2/15(ln(2))^5`

If we substitute `x = 8` now:

`F(8) = 2/15(ln(8))^5 - 2/15(ln(2))^5 approx 5.163`