Single Variable Calculus

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Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 12

Expert Answers

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Evaluate $\displaystyle \int p^5 \ln p dp$

If we let $u = \ln p$ and $dv = p^5 d_p$, then

$\displaystyle du = \frac{d_p}{p} \text{ and } v = \int p^5 d_p = \frac{p^6}{6}$


$ \begin{equation} \begin{aligned} \int p^5 \ln p d_p &= uv - \int v du = \frac{p^6}{6} \ln p - \int \frac{p^6}{6} \left( \frac{d_p}{} \right)\\ \\ &= \frac{p^6}{6} \ln p - \frac{1}{6} \int p^6 d_p\\ \\ &= \frac{p^6}{6} \ln p - \frac{1}{6} \left[ \frac{p^6}{6} \right] + c\\ \\ &= \frac{p^6}{6} \left[ \ln p - \frac{1}{6} \right] + c \end{aligned} \end{equation} $

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