evaluate the integral using integration by parts with the indicated choices of u and dv. (use c \n∫ x^8…

Explanation:

Step1: Find $du$ and $v$

Differentiate $u = \ln(x)$ to get $du=\frac{1}{x}dx$. Integrate $dv = x^{8}dx$ to get $v=\frac{1}{9}x^{9}$.

Step2: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Substitute $u$, $v$, $du$ into the formula: $\int x^{8}\ln(x)dx=\frac{1}{9}x^{9}\ln(x)-\int\frac{1}{9}x^{9}\cdot\frac{1}{x}dx$.

Step3: Simplify the new integral

$\int\frac{1}{9}x^{9}\cdot\frac{1}{x}dx=\frac{1}{9}\int x^{8}dx$.

Step4: Integrate $x^{8}$

$\frac{1}{9}\int x^{8}dx=\frac{1}{9}\cdot\frac{1}{9}x^{9}+C=\frac{1}{81}x^{9}+C$.

Step5: Write the final result

$\int x^{8}\ln(x)dx=\frac{1}{9}x^{9}\ln(x)-\frac{1}{81}x^{9}+C$.

Answer:

$\frac{1}{9}x^{9}\ln(x)-\frac{1}{81}x^{9}+C$