/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Evaluate the integral. $$ \i... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 13

Evaluate the integral. $$ \int \ln (3 x-2) d x $$

Short Answer

Expert verified
The integral is \( x \ln(3x-2) - x + \frac{2}{3} \ln|3x-2| + C \).

Step by step solution

01

Choose a Method

To evaluate the integral \( \int \ln (3x - 2) \, dx \), we will use integration by parts. This method is useful for integrals involving a product of functions, one of which is easily differentiated.
02

Recall Integration by Parts Formula

Integration by parts formula is given by:\[\int u \, dv = uv - \int v \, du\]We need to identify \( u \) and \( dv \) for the function \( \ln (3x - 2) \).
03

Identify \( u \) and \( dv \)

Let \( u = \ln(3x - 2) \) and \( dv = dx \). This choice is strategic since \( \ln(3x-2) \) is easily differentiable and \( dx \) is easily integrable.
04

Differentiate \( u \) and Integrate \( dv \)

Differentiate \( u \) to find \( du \):\[ du = \frac{1}{3x - 2} \cdot 3 \, dx = \frac{3}{3x - 2} \, dx \]Integrate \( dv \) to find \( v \):\[ v = x \]
05

Substitute into Integration by Parts Formula

Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:\[\int \ln (3x - 2) \, dx = x \ln(3x-2) - \int x \left( \frac{3}{3x - 2} \right) \, dx\]
06

Simplify the Remaining Integral

First, simplify the integral:\[ \int x \cdot \frac{3}{3x - 2} \, dx \]This simplifies algebraically to:\[ 3 \int \frac{x}{3x - 2} \, dx \]
07

Integrate Remaining Integral

Use substitution method for the integral \( 3 \int \frac{x}{3x - 2} \, dx \).Let \( v = 3x - 2 \), then \( dv = 3 \, dx \), thus, \( dx = \frac{dv}{3} \). Substitute into the integral:\[ \int \frac{x}{v} \cdot \frac{dv}{3} \]Express \( x \) in terms of \( v \): \((x = \frac{v+2}{3})\) and substitute:\[ \int \frac{v + 2}{3v} \cdot \frac{dv}{3} \]This can then be integrated directly.
08

Evaluate the Sub Integral and Final Expression

Sub-integrating, factor out the constants and integrate:\[\int \left(\frac{1}{3} + \frac{2}{3v}\right) \, dv = \frac{1}{3}v + \frac{2}{3} \ln|v|\]Convert back to \( x \):\[ = \frac{1}{3}(3x - 2) + \frac{2}{3} \ln|3x - 2|\]Combine results with Step 5:\[ x \ln(3x-2) - \frac{1}{3}(3x - 2) - \frac{2}{3} \ln|3x-2|+ C\]

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Key Concepts

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

Natural Logarithm
The natural logarithm, often written as \( \ln(x) \), is a fundamental logarithmic function in calculus. It is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828, an important mathematical constant. The function \( \ln(x) \) is specifically useful for describing growth processes such as population growth and compound interest.
Under globalization:
  • The derivative of \( \ln(x) \) is \( \frac{1}{x} \). This property is key in calculus since it makes differentiation straightforward.
  • The natural logarithm is the inverse function of the exponential function. This means that \( e^{\ln(x)} = x \) for \( x > 0 \).
  • It is essential for solving integrals involving a natural logarithm as its derivative simplifies complex functions.
The integration of a function involving a natural logarithm requires particular techniques, often those that include simplification of derivatives and leveraging logarithmic identities for ease of calculation.
Integration Techniques
Integration techniques are strategies used to simplify and solve integrals in calculus. Given the integral\[\int \ln(3x - 2) \, dx\]integration by parts is one common technique employed.
  • Integration by Parts: This technique is derived from the product rule for differentiation, expressed as \( \int u \, dv = uv - \int v \, du \). It is particularly useful when dealing with the product of functions, where one is easy to differentiate, like \( \ln(3x-2) \) being \( u \).
  • Substitution: Sometimes it might be necessary to change variables to simplify an integral, as was done in the exercise to manage the expression \( \frac{x}{3x-2} \).
  • Other techniques include partial fraction decomposition or trigonometric identities, although not needed here, are good to recognize within the toolbox of integration skills.
Understanding these techniques helps in tackling a variety of integrals beyond simple functions.
Calculus Integration Methods
Calculus integration methods encompass a wide range of approaches to find antiderivatives. These methods are pivotal in solving problems like evaluating the integral of \( \ln(3x - 2) \), concretely using integration by parts.
  • Direct Integration: Often the fastest method if the integrand is a straightforward function that fits basic integration rules.
  • Integration by Parts: Considered when direct integration isn't straightforward, particularly for function products where rearranging the integral yields simpler sub-integrals.
  • Substitution: Applies well when changing variables simplifies the integral, as it was used in the exercise by introducing \( v = 3x - 2 \) to simplify \( \frac{x}{3x - 2} \).
Each calculus integration method serves a purpose depending on the form of the integrand, allowing for flexibility and problem-solving simplicity. Mastery of these methods is essential for advanced calculus work and real-world applications involving calculus.

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Most popular questions from this chapter

Approximate the integral using Simpson's rule \(S_{10}\) and compare your answer to that produced by a calculating utility with a numerical integration capability. Express your answers to at least four decimal places. $$ \int_{-1}^{1} e^{-x^{2}} d x $$

Determine whether the statement is true or false. Explain your answer. Simpson's rule approximation \(S_{50}\) for \(\int_{a}^{b} f(x) d x\) corresponds to \(\int_{a}^{b} q(x) d x\), where the graph of \(q\) is composed of 25 parabolic segments joined at points on the graph of \(f\).

Find the volume of the solid generated when the region enclosed by \(y=\tan x, y=1\), and \(x=0\) is revolved about the \(x\) -axis.

The exact value of the given integral is \(\pi\) (verify). Approximate the integral using (a) the midpoint approximation \(M_{10},(\mathrm{~b})\) the trapezoidal approximation \(T_{10}\), and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). Approximate the absolute error and express your answers to at least four decimal places. $$ \int_{0}^{3} \frac{4}{9} \sqrt{9-x^{2}} d x $$

Evaluate the integral. $$ \int_{0}^{\pi / 6} \sec ^{3} 2 \theta \tan 2 \theta d \theta $$
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