Question

section 3.6: additional in
score: 40/60 answered: 4/6
progress saved done
question 5
compute the given integral.
$\int x^{7}\ln xdx=$

Explanation:

Step1: Use integration by parts formula

The integration by parts formula is \(\int u\mathrm{d}v=uv - \int v\mathrm{d}u\). Let \(u = \ln x\), then \(\mathrm{d}u=\frac{1}{x}\mathrm{d}x\). Let \(\mathrm{d}v=x^{7}\mathrm{d}x\), then \(v=\frac{x^{8}}{8}\) (using the power rule \(\int x^{n}\mathrm{d}x=\frac{x^{n + 1}}{n+1}+C,n
eq - 1\)).

Step2: Apply the integration by parts formula

$$
LATEXBLOCK0
$$

Step3: Integrate the remaining integral

Using the power rule \(\int x^{n}\mathrm{d}x=\frac{x^{n + 1}}{n+1}+C\) for \(n = 7\), we have \(\int x^{7}\mathrm{d}x=\frac{x^{8}}{8}+C\). So \(\frac{1}{8}\int x^{7}\mathrm{d}x=\frac{x^{8}}{64}+C\).

Answer:

\(\frac{x^{8}}{8}\ln x-\frac{x^{8}}{64}+C\)