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Magazine Chapter 7: Problem 12
\(3-32\) Evaluate the integral. $$\int p^{5} \ln p d p$$
Short Answer
Expert verified
\( \frac{p^6}{6} \ln p - \frac{p^6}{36} + C \)
Step by step solution
01
Identify the Method
Recognize that the integral \( \int p^{5} \ln p \, dp \) requires the integration by parts method. Integration by parts formula is \( \int u \, dv = uv - \int v \, du \).
02
Choose \( u \) and \( dv \)
Let \( u = \ln p \) and \( dv = p^5 \, dp \). Then, compute \( du \) and \( v \).
03
Differentiate \( u \)
Differentiate \( u = \ln p \) to get \( du = \frac{1}{p} \, dp \).
04
Integrate \( dv \)
Integrate \( dv = p^5 \, dp \) to get \( v = \frac{p^6}{6} \).
05
Apply Integration by Parts Formula
Substitute \( u, v, du, \) and \( dv \) into the integration by parts formula: \[ \int p^{5} \ln p \, dp = \frac{p^6}{6} \cdot \ln p - \int \left( \frac{p^6}{6} \cdot \frac{1}{p} \right) \, dp \].
06
Simplify the Integral
Simplify the integral: \[ \frac{1}{6} \int p^5 \, dp \].
07
Perform the Remaining Integration
Calculate \( \int p^5 \, dp \): \( \frac{1}{6} \left( \frac{p^6}{6} \right) + C = \frac{p^6}{36} + C \), where \( C \) is the constant of integration.
08
Finalize the Expression
The integral evaluates to: \[ \frac{p^6}{6} \ln p - \frac{p^6}{36} + C \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite and Indefinite Integrals
Integrals come in two main forms: definite and indefinite. Both are essential concepts in calculus and help in measuring areas under curves or solving various physical problems.
- **Indefinite Integrals**: These represent a family of functions and includes an arbitrary constant known as the constant of integration, denoted as \( C \). The indefinite integral of a function \( f(x) \) with respect to \( x \) is written as \( \int f(x) \, dx \), and it returns a new function \( F(x) \) such that the derivative \( F'(x) = f(x) \).
- **Definite Integrals**: These are used to compute the exact area under the curve from one point to another. It is presented as \( \int_{a}^{b} f(x) \, dx \) and results in a numerical value, reflecting the accumulation of the quantity \( f(x) \) from \( a \) to \( b \).
Logarithmic Integration
When dealing with logarithmic integration, the presence of a natural logarithm function \( \ln x \) often indicates that integration by parts may be beneficial.
- Integration by parts is particularly useful when a function is the product of two simpler functions, like a polynomial times a logarithm, as seen in our given integral \( \int p^5 \ln p \, dp \).
- The integration by parts formula \( \int u \, dv = uv - \int v \, du \) is used, where finding \( u \) and \( dv \) strategically simplifies the integration process.
Polynomial Functions
Polynomial functions consist of terms with variables raised to non-negative integer powers. They are among the most straightforward functions to integrate.
- To integrate a polynomial, use the power rule which states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), assuming \( n eq -1 \).
- In the exercise \( \int p^5 \ln p \, dp \), the presence of a \( p^5 \) term fits this pattern well. Hence, after decomposing the integral, integrating \( p^5 \) is straightforward using the power rule.
