Single Variable Calculus

Start Free Trial

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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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

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

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

So,

$ \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} $

Posted on

Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial

Already a member? Log in here.

Are you a teacher? Sign up now