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Chapter 4: Problem 27

Evaluate the indicated integral. $$\int \frac{2 x+3}{x+7} d x$$

Short Answer

Expert verified
The evaluated integral \( \int \frac{2x + 3}{x+7} dx = 2x - 3 \ln |x + 7| + C \)

Step by step solution

01

Rewrite the integrand as a sum of two functions

The integral can be rewritten as the sum of two simpler integrals by dividing each term in the numerator by the denominator: \( \int \frac{2x + 3}{x+7} dx = \int (\frac{2x}{x+7} + \frac{3}{x+7}) dx = \int 2 dx - \int \frac{3}{x+7} dx \)
02

Use the integral rules

The integral of the sum or difference of two functions is the sum or difference of their integrals. Also, the integral of a constant multiple of a function is the constant multiple of the integral of the function. In this case, the integral of \(2 dx = 2x\) and the integral of \( \frac{3}{7x} dx = 3 \ln |x + 7|\). So we have \(2x - 3 \ln |x + 7| + C \) where \(C\) is the constant of integration.
03

Final answer

Combine the results from the previous step to express the answer for the integral: \( \int \frac{2x + 3}{x+7} dx = 2x - 3 \ln |x + 7| + C \)

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Key Concepts

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

Integration Techniques
Integration is much like putting together pieces of a puzzle, where each piece represents a specific integration technique.
When faced with an integral like \( \int \frac{2x + 3}{x+7} dx \), the right technique can make the problem more manageable. Some common techniques include substitution, integration by parts, and, as in our exercise, partial fraction decomposition.

Each technique is chosen based on the form of the integrand (the function you are integrating). For instance, partial fraction decomposition is particularly useful when you are dealing with a rational function that can be broken into simpler fractions.
Indefinite Integral
An indefinite integral refers to the general form of an antiderivative without specifying the limits of integration.
Essentially, this represents a family of functions that differs by a constant. The integral \( \int \frac{2x + 3}{x+7} dx \) is indefinite, meaning our answer will include an arbitrary constant \( C \).

This constant represents all possible vertical shifts of the antiderivative; thus, indefinite integrals are essential for understanding how functions can be reconstructed from their rates of change.
Partial Fractions
Partial fractions are a type of 'algebraic surgery' used on rational functions (where we have a polynomial divided by another polynomial). The goal is to break down a complicated fraction into simpler pieces that are easier to integrate.

In our exercise, we split \( \frac{2x + 3}{x+7} \) into \( \frac{2x}{x+7} \) and \( \frac{3}{x+7} \) before integrating. By doing so, we transform a seemingly complex integral into manageable parts, each of which can often be tackled with basic integration rules.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions, and they appear frequently in integration, especially when dealing with integrals of fractions.

In the given problem, after applying partial fractions, we encountered the integral \( \int \frac{3}{x+7} dx \), which naturally leads to a logarithmic function. The antiderivative of \( \frac{1}{x} \) is \( \ln|x| \), so for \( \frac{3}{x+7} \) we get \( 3 \ln|x+7| \). This beautifully illustrates how integrals give us natural logarithms, linking algebraic expressions to the growth patterns described by logs.

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Most popular questions from this chapter

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