let F(x) be an antiderivative of (lnx)^3 / x. If F(1) = 0, then F(9) = ?

Question

please show all work! thank you.

Answer 1

We know that F(x) = ∫ln³ x/x dx + C. So we need to find that integral, solve for the constant of integration, and then plug 9 into the resulting formula. First, the integral:

∫ln³ x/x dx
Let u=ln³ x, v=ln x, du=3 ln² x/x dx, dv = 1/x dx. Integrating by parts:
∫ln³/x dx = ln^4 x - ∫3 ln³ x/x dx + C
Adding 3∫ln³ x/x dx to both sides:
4∫ln³/x dx = ln^4 x + C
Dividing by 4:
∫ln³/x dx = ln^4 x + C

Now, solving for C:

F(1) = ln^4 1 + C
F(1) = 0^4 + C
F(1) = C
But F(1) = 0, so C=0.

Now, solving for F(9):

F(9) = ln^4 9 = (ln 9)^4 ≈ 23.30761

Answer 2

The answer is not 23.30761 but 5.8269. The antiderivative of ln^3x/x is ln^4x/4 +C not just ln^4x+C