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Chapter 7: Problem 29

Evaluate the following integrals. $$\int \frac{x+2}{x+4} d x$$

Short Answer

Expert verified
Based on the step-by-step solution provided above, the final result for the integral $\int \frac{x + 2}{x + 4} dx$ is $(x + 4) - 2\ln{|x + 4|} + C$, where C is the constant of integration.

Step by step solution

01

Perform substitution

Let's substitute \(u = x + 4\). Now we need to find the derivative of u with respect to x to substitute it in the integral: $$\frac{d u}{d x} = \frac{d (x + 4)}{d x} = 1$$ So, \(dx = du\). Now, we need to change the limits of the integral and expression inside it according to the new variable u: $$\int \frac{x + 2}{x + 4} dx = \int\frac{u-2}{u} du$$ The integral now becomes: $$\int \frac{u - 2}{u} du$$
02

Split the fraction

Now, we can split the fraction inside the integral to make it easier to integrate: $$\int \frac{u - 2}{u} du = \int \left(\frac{u}{u} - \frac{2}{u}\right) du$$ Which simplifies to: $$\int (1 - 2\cdot \frac{1}{u}) du$$
03

Integrate

Now, we can easily integrate each term: $$\int (1 - 2\cdot \frac{1}{u}) du = \int 1 du - 2 \int \frac{1}{u} du$$ The first integral, \(\int 1 du\), gives us \(u\). The second integral, \(\int \frac{1}{u} du\), is a well-known integral, and its result is \(\ln{|u|}\). Thus, we have: $$\int (1 - 2\cdot \frac{1}{u}) du = u - 2\ln{|u|} + C$$ Where C is the constant of integration.
04

Substitute back the original variable

Now, we need to substitute back x in terms of u: $$u - 2\ln{|u|} + C = (x + 4) - 2\ln{|x + 4|} + C$$ Therefore, the final result is: $$\int \frac{x + 2}{x + 4} dx = (x + 4) - 2\ln{|x + 4|} + C$$

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Key Concepts

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

Substitution Method
Substitution method is a powerful technique used to simplify complex integrals. Think of it as changing the 'view' of the problem. We substitute a part of the integral with a new variable, making it easier to solve.
Start by choosing a substitution that simplifies the integral. For example, if you have an expression like \(x + 4\), you can let \(u = x + 4\).
  • This simplifies the integral because you'll transform the variables involved.
  • Calculate \(dx\) by differentiating \(u\) with respect to \(x\).
After substituting, the integral may become more straightforward. In our example, with \(u = x + 4\), the integral transforms from \(\int \frac{x + 2}{x + 4} dx\) to \(\int \frac{u - 2}{u} du\).
Splitting, or breaking down, complex fractions often makes integration easier.
Definite and Indefinite Integrals
Integrals can be either definite or indefinite, and understanding the difference is crucial. Indefinite integrals represent a family of functions and are expressed with a constant \(C\), whereas definite integrals calculate the net area under a curve.
To solve an indefinite integral, simply integrate the function without setting any limits. For example, in our indefinite integral \(\int (1 - 2\frac{1}{u}) du\), the result is a function plus \(C\).
Definite integrals, on the other hand, have limits and calculate the net accumulation between those bounds. They are typically notated with upper and lower limits on the integral sign. Although our example focused on indefinite integrals, understanding both concepts helps in various applications like calculating areas or volumes.
Logarithmic Integration
Logarithms often appear in integration, especially with fractional forms. In our example, when we encounter integrands like \(\int \frac{1}{u} du\), the result is a logarithmic function.
  • The integral \(\int \frac{1}{u} du\) is one of the basic logarithmic integrals and equals \ln{|u|}\.
  • Logarithmic results frequently appear in integrals where the denominator is the derivative of the numerator, such as \(\int \frac{1}{u} du\), which simplifies to \ln{|u|}\.
When integrating, always remember the constant \(C\) for indefinite integrals. This indicates the family of solutions and is crucial for problems involving initial conditions or specific values.

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