use integration by parts to evaluate the following integral assuming that x > 0. ∫x^5 ln(x) dx

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:

$\frac{x^{6}}{6}\ln(x)-\frac{x^{6}}{36}+C$