7.1 Integration By Parts/12: Difference between revisions

Revision as of 03:29, 29 November 2022

Evaluate the integral.

${\begin{aligned}\int p^{5}ln(p)dp\end{aligned}}$

${\begin{aligned}u=ln(p)\quad \quad dv=p^{5}dp\\[2ex]du={\frac {1}{p}}\quad \quad \quad v={\frac {1}{6}}p^{6}\\[2ex]\end{aligned}}$

${\begin{aligned}\int p^{5}ln(p)dp={\frac {1}{6}}p^{6}ln(p)-\int p^{6}{\frac {1}{p}}dp={\frac {1}{6}}p^{6}ln(p)-{\frac {1}{36}}p^{6}+C\end{aligned}}$

@@ Line 2: / Line 2: @@
 <math>
+\begin{align}
 \int p^5 ln (p)dp
+\end{align}
+</math>
+<math>
+\begin{align}
+u= ln(p) \quad \quad  dv=p^5dp \\[2ex]
+du = \frac{1}{p} \quad \quad \quad v= \frac{1}{6}p^6  \\[2ex]
+\end{align}
+</math>
+<math>
+\begin{align}
+\int p^5 ln (p)dp = \frac{1}{6}p^6 ln(p) - \int p^6 \frac{1}{p} dp = \frac{1}{6}p^6 ln(p)-\frac{1}{36}p^6+C
+\end{align}
 </math>