use integration by parts to evaluate the foll...

# Explanation: ## Step1: Choose $u$ and $dv$ Let $u = \ln(x)$ and $dv=x^{5}dx$. ## Step2: Find $du$ and $v$ Differentiate $u$: $du=\frac{1}{x}dx$. Integrate $dv$: $v=\int x^{5}dx=\frac{x^{6}}{6}$. ## Step3: 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: $\int x^{5}\ln(x)dx=\frac{x^{6}}{6}\ln(x)-\int\frac{x^{6}}{6}\cdot\frac{1}{x}dx$. ## Step4: Simplify the new integral $\int\frac{x^{6}}{6}\cdot\frac{1}{x}dx=\frac{1}{6}\int x^{5}dx$. ## Step5: Integrate the simplified integral $\frac{1}{6}\int x^{5}dx=\frac{1}{6}\cdot\frac{x^{6}}{6}+C=\frac{x^{6}}{36}+C$. # Answer: $\frac{x^{6}}{6}\ln(x)-\frac{x^{6}}{36}+C$