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Chapter 6: Problem 41

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) $$ \int_{1}^{3} x^{2} \ln x d x $$

Short Answer

Expert verified
The value of the definite integral is \( 9 \ln 3 - \frac{26}{9} \).

Step by step solution

01

Identify Parts for Integration by Parts

For integration by parts, we use the formula \( \int u \, dv = uv - \int v \, du \). We need to identify \( u \) and \( dv \). Let \( u = \ln x \) and \( dv = x^2 \, dx \).
02

Differentiate and Integrate

Differentiate \( u \) to find \( du \) and integrate \( dv \) to find \( v \).- \( du = \frac{1}{x} \, dx \)- \( v = \frac{x^3}{3} \) (since \( \int x^2 \, dx = \frac{x^3}{3} \))
03

Apply Integration by Parts Formula

Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:\[ \int x^2 \ln x \, dx = \left( \ln x \right) \left( \frac{x^3}{3} \right) - \int \left( \frac{x^3}{3} \right) \left( \frac{1}{x} \right) \, dx \]
04

Simplify and Integrate Remaining Expression

Simplify the integral expression:\[ = \frac{x^3}{3} \ln x - \frac{1}{3} \int x^2 \, dx \]Evaluate the new integral:\[ \frac{1}{3} \int x^2 \, dx = \frac{1}{3} \left( \frac{x^3}{3} \right) = \frac{x^3}{9} \]
05

Evaluate Definite Integral

Substitute and evaluate from 1 to 3:\[ \left[ \frac{x^3}{3} \ln x - \frac{x^3}{9} \right]_1^3 \]
06

Evaluate Limits

Evaluate the expression at the upper limit 3:- At \( x = 3 \): \( \frac{27}{3} \ln 3 - \frac{27}{9} = 9 \ln 3 - 3 \)Evaluate the expression at the lower limit 1:- At \( x = 1 \): \( \frac{1^3}{3} \ln 1 - \frac{1^3}{9} = 0 - \frac{1}{9} = -\frac{1}{9} \) (since \( \ln 1 = 0 \))
07

Calculate the Definite Integral

Subtract values from the upper and lower limits:\[ (9 \ln 3 - 3) - \left(-\frac{1}{9}\right) = 9 \ln 3 - 3 + \frac{1}{9} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
A definite integral represents the signed area under a curve between two bounds on the x-axis, in this case from 1 to 3. It's written with an integral sign with the lower limit as the bottom number and the upper limit as the top number, \(\int_{1}^{3} f(x) \, dx\). Unlike indefinite integrals, definite integrals give a specific number as a result. This number represents the accumulated value, often thought of as area, from the start to the end of the interval.

Main features include:
  • Specified limits: The limits (1 and 3 in this case) determine the interval over which we're integrating.
  • An exact value: Definite integrals yield a numerical value after evaluation, which differs from indefinite integrals that yield a family of functions.
  • Signed area: Depending on the function and interval, the result might be positive or negative, reflecting the concept of signed area in calculus.
In our specific exercise, the definite integral involves a function with a natural logarithmic function, making the problem interesting due to its non-polynomial nature.
Natural Logarithm
The natural logarithm, noted as \( \ln x \) is the logarithm to the base of the mathematical constant \(e\), approximately equal to 2.71828. The natural logarithm is a particular logarithmic function that is prevalent in calculus due to its mathematical properties and how naturally it arises in several growth and decay processes.

Key aspects of natural logarithms include:
  • Derivative: The derivative of \( \ln x \) is straightforward, being \( \frac{1}{x} \), which is often useful in integration by parts as it simplifies parts of the expression.
  • Logarithm rules: Using properties such as \( \ln(ab) = \ln a + \ln b \) and \( \ln(a/b) = \ln a - \ln b \) helps simplify complex logarithmic expressions.
  • Connection to exponentials: The natural logarithm is the inverse function of the exponential function \( e^x \), thus \( \ln(e^x) = x \).
In our exercise, the natural logarithm \( \ln x \) was chosen to be \( u \) in the integration by parts formula due to its properties, which simplify differentiation and integration steps.
Calculus Problem Solving
Calculus problem solving requires an array of techniques to tackle various problems, including definite integrals with unconventional functions such as \( x^2 \ln x \). A firm understanding of calculus concepts allows students to handle these challenges systematically.

Steps for solving such an integral using integration by parts involve:
  • Identifying functions: Choosing appropriate functions for \( u \) and \( dv \), such as setting \( u = \ln x \) and \( dv = x^2 \, dx \), simplifies the differentiation and integration process.
  • Differentiation and integration: We compute the derivative of \( u \) and the integral of \( dv \, \) which are crucial for correctly applying the integration by parts formula.
  • Simplification: Breaking down and simplifying the integral repeatedly by integrating smaller parts helps in achieving the final result of a calculus problem.
  • Limit evaluation: Applying and evaluating limits at the final step ensures precision, converting the indefinite integral into a definite one.
By sequentially applying these steps, students can master calculus problems, improving their logical reasoning and computational skills. Through practice, students develop confidence in solving tricky integrals with techniques like integration by parts.

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Most popular questions from this chapter

$$ \text { Use integration by parts to find each integral. } $$ $$ \int \frac{x+1}{e^{3 x}} d x $$

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$$ \text { Use integration by parts to find each integral. } $$ $$ \int(x+2)(x-5)^{5} d x $$
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