use integration by parts to evaluate the following integral assuming that x > 0. ∫x⁴ln(x)dx

Explanation:

Step1: Choose $u$ and $dv$

Let $u = \ln(x)$ and $dv=x^{4}dx$.

Step2: Find $du$ and $v$

Differentiate $u$: $du=\frac{1}{x}dx$. Integrate $dv$: $v=\frac{x^{5}}{5}$.

Step3: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. So $\int x^{4}\ln(x)dx=\frac{x^{5}}{5}\ln(x)-\int\frac{x^{5}}{5}\cdot\frac{1}{x}dx$.

Step4: Simplify the new integral

$\int\frac{x^{5}}{5}\cdot\frac{1}{x}dx=\frac{1}{5}\int x^{4}dx$.

Step5: Integrate $x^{4}$

$\frac{1}{5}\int x^{4}dx=\frac{1}{5}\cdot\frac{x^{5}}{5}+C=\frac{x^{5}}{25}+C$.

Answer:

$\frac{x^{5}}{5}\ln(x)-\frac{x^{5}}{25}+C$