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Magazine Chapter 6: Problem 60
Evaluate the indefinite integral. \(\int \frac{7}{3 x+2} d x\)
Short Answer
Expert verified
\( \frac{7}{3} \ln |3x + 2| + C \)
Step by step solution
01
Identify the Type of Integral
The integral \( \int \frac{7}{3x + 2} \, dx \) is a rational function integral. This means we can apply a substitution, or algebraic manipulation to solve it.
02
Choose a Substitution
We will let \( u = 3x + 2 \). This substitution simplifies the integral, focusing on the expression in the denominator.
03
Determine \( du \)
Differentiate \( u = 3x + 2 \) with respect to \( x \). This gives us \( \frac{du}{dx} = 3 \), or \( du = 3 \, dx \).
04
Solve for \( dx \)
Rearrange the equation \( du = 3 \, dx \) to find \( dx \): \( dx = \frac{du}{3} \).
05
Substitute in the Integral
Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{7}{u} \cdot \frac{1}{3} \, du \].
06
Simplify the Integral
Factor out constants from the integral: \( \int \frac{7}{3u} \, du = \frac{7}{3} \int \frac{1}{u} \, du \).
07
Evaluate the Integral
The integral \( \int \frac{1}{u} \, du \) is a standard form, resulting in \( \ln|u| + C \). Substitute this back into our expression: \( \frac{7}{3} \ln|u| + C \).
08
Substitute Back in Terms of \( x \)
Since \( u = 3x + 2 \), substitute \( u \) back with \( 3x + 2 \): \( \frac{7}{3} \ln|3x + 2| + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Substitution
Integration by substitution is a technique used to solve integrals by changing variables. This makes the integral easier to evaluate. In this approach, we substitute a part of the integral with a new variable. We also adjust the differential accordingly. The objective is to transform the integral into a simpler form.
- First, choose the substitution. Look for a function within the integral that can be simplified. In this case, we use the linear expression in the denominator, choosing \( u = 3x + 2 \).
- Then, find the derivative of the substitution with respect to \( x \), leading to \( du = 3 \, dx \).
- Then, express \( dx \) in terms of \( du \), so \( dx = \frac{du}{3} \). Substitute \( u \) and \( dx \) back into the integral.
Rational Function Integral
A rational function is any function that can be expressed as the ratio of two polynomial functions. In our exercise, the integral has the form \( \frac{7}{3x+2} \).
- Since a rational function is involved, the goal is to simplify it so that the integration process is easier.
- Integration by substitution often helps when dealing with linear denominators. With substitutions, rational functions become simpler to integrate.
- The new integral \( \int \frac{7}{u} \cdot \frac{1}{3} \, du \) is a much simpler rational expression.
Natural Logarithm
The natural logarithm, denoted \( \ln \), is the logarithm to the base \( e \). In calculus, integrating certain rational functions often results in natural logarithms.
- When the integral has the form \( \int \frac{1}{u} \, du \), the result is \( \ln|u| + C \), where \( C \) is the constant of integration.
- It's crucial to remember the absolute value around \( u \), as the natural logarithm is defined only for positive values.
Antiderivative
The antiderivative is a function whose derivative is the original function. Calculating antiderivatives is the reverse of differentiation. When we compute the indefinite integral of a function, we are essentially finding its antiderivative.
- Given the problem, \( \int \frac{7}{3x+2} \, dx \), finding the antiderivative involves a process that reverses differentiation on the function.
- By using substitution, the integral is transformed and solved into a form involving \( \ln|3x+2| \).
- This result, \( \frac{7}{3} \ln|3x+2| + C \), represents the antiderivative of the original rational function.
