Let F(x) be an antiderivative of 2(ln⁡x)43x, If F(2)=0, then F(8)=?

"Let F(x) be an antiderivative of 2(ln⁡x)43x, If..." was answered by our community

Taniyah Hartman

2023-01-14

Let F(x) be an antiderivative of

\frac{2 {(\ln x)}^{4}}{3 x}

, If F(2)=0, then F(8)=?

Mikayla Cox

Beginner2023-01-15Added 15 answers

We have:

f (x) = \frac{2 {(\ln (x))}^{4}}{3 x}

We need integrate. Let

τ \equiv \ln (x)

F (x) = \int \frac{2 τ^{4}}{3} d τ

F (x) = \frac{2}{15} τ^{5} + C

F (x) = \frac{2}{15} {(\ln (x))}^{5} + C

where C is a randomly chosen integration constant.
We have more information, that

F (2) = 0

F (2) = \frac{2}{15} {(\ln (2))}^{5} + C = 0

C = - \frac{2}{15} {(\ln (2))}^{5}

Then, we find:

F (x) = \frac{2}{15} {(\ln (x))}^{5} - \frac{2}{15} {(\ln (2))}^{5}

If we substitute

x = 8

now:

F (8) = \frac{2}{15} {(\ln (8))}^{5} - \frac{2}{15} {(\ln (2))}^{5} \approx 5.163

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