Question

evaluate the integral.
int p^{5}ln p dp
enhanced feedback
please try again. recall that (int u dv = uv-int v du).
try letting (u = ln p) and (dv = p^{n}dp).

Explanation:

Step1: Apply integration - by - parts formula

Let $u = \ln p$ and $dv=p^{5}dp$. Then $du=\frac{1}{p}dp$ and $v=\frac{p^{6}}{6}$ (using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C,n
eq - 1$).

Step2: Substitute into integration - by - parts formula $\int u\;dv=uv-\int v\;du$

We have $\int p^{5}\ln p\;dp=\frac{p^{6}}{6}\ln p-\int\frac{p^{6}}{6}\cdot\frac{1}{p}dp$.

Step3: Simplify the second integral

$\int\frac{p^{6}}{6}\cdot\frac{1}{p}dp=\frac{1}{6}\int p^{5}dp$.

Step4: Integrate $p^{5}$

Using the power - rule for integration again, $\frac{1}{6}\int p^{5}dp=\frac{1}{6}\cdot\frac{p^{6}}{6}+C=\frac{p^{6}}{36}+C$.

Answer:

$\frac{p^{6}}{6}\ln p-\frac{p^{6}}{36}+C$