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Chapter 8: Problem 24

Finding an Indefinite Integral In Exercises \(15-46\) , find the indefinite integral. $$ \int \frac{3 x}{x+4} d x $$

Short Answer

Expert verified
The indefinite integral of \( \int \frac{3 x}{x+4} d x \) is \( 3(x + 4) - 12 \ln |x + 4| + C \).

Step by step solution

01

Identify the function in the denominator

In the integral \( \int \frac{3 x}{x+4} d x \), the function in the denominator is \( f(x) = x + 4 \).
02

Differentiate the function

Differentiate \( f(x) \) to get \( f'(x) = 1 \). The derivative does not match the numerator (which is 3x). Therefore, the integrand does not exactly fit the \( \int \frac{f'(x)}{f(x)} dx \) form. However, this can be worked around by factoring a 3 from the numerator.
03

Factor out the 3

Factor the 3 out so the integral now looks like \( 3 \int \frac{x}{x + 4} dx \).
04

Substitution

Now make a substitution, let \( u = x + 4 \). From this, \( du = dx \). The integral then becomes \( 3 \int \frac{u - 4}{u} du \).
05

Simplify the Integral and Find the Antiderivative

The integral is now in a more workable form. Break it down into simpler fractions and solve: \( 3 \int \frac{u - 4}{u} du = 3 \int du - 3 \int \frac{4}{u} du = 3u - 12 \ln |u| + C \).
06

Back Substitute

To complete the solution, back substitute \( u = x + 4 \) into the answer: \( 3(x + 4) - 12 \ln |x + 4| + C \).

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Key Concepts

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

Integration Techniques
The process of finding indefinite integrals involves various techniques to tackle different types of functions. Each technique is useful for particular forms of integrands (the function being integrated). Here are some common integration techniques you might encounter:
  • Substitution: This is great for functions that can be rewritten as a simpler form using a new variable.
  • Partial Fractions: Ideal for integrating rational functions by breaking them into simpler fractions.
  • Integration by Parts: Useful for products of functions, using the formula derived from the product rule of differentiation.
Choosing the right technique is crucial. In the problem, we transformed the original integrand into a simpler expression, illustrating the versatility of integration techniques.
Substitution Method
The substitution method simplifies the integration process by introducing a new variable to replace a complicated expression. This is like unraveling a tangled thread into a straight line.
In our exercise, we identified the expression in the denominator, \( x + 4 \), and replaced it with a new variable \( u \), where \( u = x + 4 \). Then, we found the differential, \( du = dx \). This simplifies the integrand into a more straightforward form:
  • The integral \( \int \frac{3x}{x+4} \, dx \) becomes \( 3 \int \frac{u-4}{u} \, du \).
This substitution is the key to making the problem manageable, turning a complex integral into something simpler to evaluate.
Antiderivative
The antiderivative, or indefinite integral, is essentially the reverse process of differentiation. Finding an antiderivative involves determining the original function whose derivative yields the given function.
In our exercise, after substitution and simplification, we found the antiderivative:
  • \( 3 \int du - 3 \int \frac{4}{u} \, du = 3u - 12 \ln |u| + C \)
Here, \( 3u \) represents the antiderivative of a constant function, and \(-12 \ln |u|\) is the antiderivative of the logarithmic form \( \frac{1}{u} \). This demonstrates how integration restores the function from its derivative.
Logarithmic Integration
Logarithmic integration is used when dealing with rational functions, especially of the form \( \int \frac{1}{x} \, dx \).
This technique is essential in our example, where part of the process required integrating \( \int \frac{4}{u} \, du \), leading us to the expression \(-12 \ln |u|\).
  • The natural logarithm \( \ln |x| \) form appears frequently in results of integrals involving division by \( x \) or \( u \).
This method highlights why understanding logarithms is crucial in calculus, as they often provide the solution to complex integrals involving rational expressions.

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Most popular questions from this chapter

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