Question

$3-32$ Evaluate the integral. $$\int p^{5} \ln p d p$$

   $3-32$ Evaluate the integral.
$$\int p^{5} \ln p d p$$

Calculus Early Transcendentals

Calculus Early Transcendentals

James Stewart 6th Edition

Chapter 7, Problem 12

Instant Answer

Solved by Expert Tyler Moulton

08/13/2021

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Here, we choose $u = \ln p$ and $dv = p^5 dp$. Show more…

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$3-32$ Evaluate the integral. $$\int p^{5} \ln p d p$$

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Key Concepts

Integration by Parts

A method used to integrate products of functions by selecting parts of the integrand as u and dv, based on the formula ? u dv = uv - ? v du. This technique is particularly useful when dealing with products involving logarithmic functions, where differentiating the logarithm simplifies the expression.

Logarithmic Functions

Logarithmic functions have properties that make them suitable choices for the part u in integration by parts due to their differentiability. Recognizing that the derivative of ln p is 1/p is crucial for simplifying integrals where logarithms are multiplied by other functions.

Polynomial Integration

Polynomial integration refers to the process of integrating functions of the form p^n, for which the antiderivative is given by p^(n+1)/(n+1) when n is not equal to -1. This fundamental technique is essential when handling products of polynomial and logarithmic functions in integrals.

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