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Chapter 9: Problem 28

Determine the integrals by making appropriate substitutions. $$\int \frac{d x}{3-5 x}$$

Short Answer

Expert verified
\( -\frac{1}{5} \text{ln} |3 - 5x| + C \)

Step by step solution

01

Identify the substitution

Notice that the denominator is in the form of a linear function, so let's make a substitution to simplify it. Set \( u = 3 - 5x \).
02

Differentiate the substitution

Differentiate both sides of the equation with respect to \( x \). Therefore, \( \frac{du}{dx} = -5 \); solving for \( dx \) yields \( dx = \frac{du}{-5} \).
03

Substitute back into the integral

Substitute \( u \) and \( dx \) back into the integral. The integral becomes \( \frac{1}{-5} \times \frac{1}{u} du \).
04

Integrate

Integrate the simpler expression \( \frac{1}{-5} \times \frac{1}{u} du \). The antiderivative of \( \frac{1}{u} \) is \( \text{ln} |u| \). So, the result is \( \frac{1}{-5} \times \text{ln} |u| + C \).
05

Substitute back to x

Replace \( u \) with \( 3 - 5x \) in the answer. Thus, the final answer is \( -\frac{1}{5} \text{ln} |3 - 5x| + C \).

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Key Concepts

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

substitution method
The substitution method is a crucial integration technique that helps simplify complex integrals. Essentially, it involves changing the variables to make the integration process easier. In this exercise, we used a substitution to simplify the integral of \( \frac{1}{3-5x}dx \). By setting \( u = 3 - 5x \), we transformed the integral into a simpler form. This method is particularly useful when dealing with integrals involving functions inside another function. Remember, when making a substitution:
  • Identify a part of the integral that can be set as \( u \)
  • Differentiate \( u \) to find \( \frac{du}{dx} \)
  • Solve for \( dx \)
Breaking down the integral often simplifies it to a form that is easier to integrate.
integration techniques
Different integration techniques are designed to tackle various forms of integrals. Some common techniques include:
  • Substitution Method: Used for integrals involving a function and its derivative.
  • Integration by Parts: Useful for products of functions.
  • Partial Fraction Decomposition: Applied to rational functions.
In this exercise, we focused on the substitution method. Breaking apart the integral \( \frac{1}{3-5x}dx \) using substitution transformed it into a more manageable integral. Each technique has its own set of rules and applications, so knowing which to use and when is key to solving integrals efficiently.
antiderivative
An antiderivative is a function whose derivative is the original function. Finding the antiderivative is essentially the process of integration. In this exercise, after applying the substitution, we found the antiderivative of \( \frac{1}{u}dx \), which is \( \text{ln}|u| \). Finally, we replaced \( u \) with the original expression \( 3 - 5x \) to obtain the antiderivative of the initial integral. When computing antiderivatives, always remember to
  • Apply the correct integration technique
  • Add the constant of integration \( C \)
  • Substitute back to the original variable if necessary
This highlights the importance of understanding and being able to apply the correct integration techniques.

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Most popular questions from this chapter

Determine the integrals by making appropriate substitutions. $$\int \frac{x^{4}}{x^{5}+1} d x$$

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