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Magazine Chapter 6: Problem 7
Evaluate the indefinite integral to develop an understanding of Substitution. $$ \int \frac{1}{2 x+7} d x $$
Short Answer
Expert verified
\( \frac{1}{2} \ln|2x+7| + C \)
Step by step solution
01
Identify the Inner Function
In the integral \( \int \frac{1}{2x+7} \, dx \), the expression \( 2x+7 \) is the inner function because it is inside the denominator. This expression is what we will substitute.
02
Choose and Define the Substitution
Let \( u = 2x+7 \). This substitution simplifies the differentiation inside the integral. Differentiating both sides with respect to \( x \) gives \( \frac{du}{dx} = 2 \). Thus, \( du = 2 \, dx \), or equivalently \( dx = \frac{1}{2} \, du \).
03
Substitute and Modify the Integral
Substitute \( u \) and \( dx \) in the integral, which gives:\[ \int \frac{1}{u} \frac{1}{2} \, du \]This can be rewritten by factoring out the constant:\[ \frac{1}{2} \int \frac{1}{u} \, du \]
04
Integrate Using the Basic Logarithmic Integral
The integral of \( \frac{1}{u} \) is a basic logarithmic integral, which evaluates to:\[ \int \frac{1}{u} \, du = \ln|u| + C \]Thus, our integral becomes:\[ \frac{1}{2} (\ln|u| + C) \]Simplifying gives:\[ \frac{1}{2} \ln|u| + C \]
05
Back-Substitute to Original Variables
Replace \( u \) back with the original expression \( 2x+7 \):\[ \frac{1}{2} \ln|2x+7| + C \]
06
Finalize the Solution
The indefinite integral of \( \int \frac{1}{2x+7} \, dx \) is:\[ \frac{1}{2} \ln|2x+7| + C \]This is the final answer, where \( C \) is the constant of integration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
Substitution is a crucial technique in calculus that simplifies the integration process. Imagine you're trying to integrate a complicated function. The substitution method allows you to replace part of the function with a single variable, making the problem much easier. This method is particularly useful when dealing with indefinite integrals.
Let's break it down:
Let's break it down:
- Identify the Inner Function: The first step is to recognize the complicated part of your integrand. In our example, we identified \( 2x+7 \) as the inner function because it affects the complexity by being inside the denominator.
- Choose a Substitution: We assign a new variable, at this point in our example we said \( u = 2x+7 \). This substitution will take the place of the complex expression and simplify the derivative.
- Find the Differential: When you differentiate \( u = 2x+7 \), you get \( \frac{du}{dx} = 2 \). Solving for \( dx \) gives us \( dx = \frac{1}{2} \, du \). This is essential for adjusting the differentials in the integral.
Logarithmic Integration
Logarithmic integration comes into play when we encounter integrals in the form of \( \int \frac{1}{x} \, dx \). This form leads to the natural logarithmic function in calculus. In our example, after substitution, the integral \( \int \frac{1}{u} \, du \) is a classic logarithmic integral.
Here's how you can handle it:
Here's how you can handle it:
- Recognize the Form: Once the substitution has been made, check if the integral has become \( \int \frac{1}{u} \, du \). This indicates that logarithmic integration will be useful.
- Apply the Logarithmic Rule: The integral \( \int \frac{1}{u} \, du \) evaluates to \( \ln|u| + C \). This rule is derived from the integral of \( \frac{1}{x} \), leading directly to the natural logarithm of the absolute value.
Constant of Integration
In calculus, the constant of integration \( C \) is an essential part of evaluating indefinite integrals. It represents an infinite family of solutions, as the process of integration loses any constant that was originally present in the derivative.
Here's a closer look:
Here's a closer look:
- Why It's Important: After integration, you might wonder why we add \( C \). This constant ensures that the solution covers all possible antiderivatives of a given function.
- Indefinite Integrals: Unlike definite integrals, which yield a specific number, indefinite integrals result in a general form, \( F(x) + C \), where \( F(x) \) represents an antiderivative.
- Finalizing Your Solution: Always include \( C \) when writing the final answer. For example, our result was \( \frac{1}{2} \ln|2x+7| + C \), reflecting all the possible shifts vertically that the integral's antiderivative can have.
