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Magazine Chapter 28: Problem 4
Integrate each of the given functions. $$\int \frac{d x}{1-4 x}$$
Short Answer
Expert verified
\(-\frac{1}{4} \ln |1 - 4x| + C\)
Step by step solution
01
Recognize the Basic Form
The integral \( \int \frac{dx}{1-4x} \) can be recognized as a simple substitution integral. This is due to its form, which resembles \( \int \frac{dx}{a+bx} \), a standard form in integration.
02
Choose the Substitution
For the integral \( \int \frac{dx}{1-4x} \), we perform a substitution to simplify it. Let \( u = 1 - 4x \). Hence, \( du = -4 \, dx \), or equivalently \( dx = -\frac{1}{4} \, du \).
03
Substitute in the Integral
Substitute \( u = 1 - 4x \) and \( dx = -\frac{1}{4} \, du \) into the integral. The expression becomes:\[ \int \frac{dx}{1-4x} = \int \frac{-\frac{1}{4} \, du}{u} = -\frac{1}{4} \int \frac{du}{u}. \]
04
Integrate the Simplified Expression
The integral \( \int \frac{du}{u} \) is a standard logarithmic form, giving \( \ln |u| + C \) where \( C \) is the integration constant. Thus:\[ -\frac{1}{4} \int \frac{du}{u} = -\frac{1}{4} \ln |u| + C. \]
05
Back Substitute for the Original Variable
Now, replace \( u \) with the original expression in terms of \( x \). Since \( u = 1 - 4x \), substituting back gives:\[ -\frac{1}{4} \ln |1 - 4x| + C. \]
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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 useful technique in integration for handling complex integrals. This approach involves choosing an appropriate substitution to simplify the integral, essentially transforming it into a simpler form that is easier to integrate. The basic idea is to substitute part of the integrand (the function being integrated) with a new variable.
- The first step is to identify a part of the function that can be substituted to simplify the integral. In the given exercise, we used the substitution \( u = 1 - 4x \).
- This substitution allows us to express \( dx \) in terms of \( du \). Here, \( du = -4 \, dx \) leading to \( dx = -\frac{1}{4} \, du \).
- Finally, substitute back into the integral to transform it: \( \int \frac{dx}{1-4x} \) becomes \( \int \frac{-\frac{1}{4} \, du}{u} \).
Logarithmic Integration
Logarithmic integration is a crucial technique when dealing with integrals of the form \( \int \frac{1}{x} \, dx \). The result of such an integral is the natural logarithm of the absolute value of \( x \), namely \( \ln |x| \).
In our specific exercise, once we carried out the substitution method, we ended up with the integral:
\[ -\frac{1}{4} \int \frac{du}{u} \].
In our specific exercise, once we carried out the substitution method, we ended up with the integral:
\[ -\frac{1}{4} \int \frac{du}{u} \].
- This integral follows the property \( \int \frac{du}{u} = \ln |u| + C \), where \( C \) is the constant of integration.
- Thus, integrating \( \int \frac{du}{u} \) results in \( \ln |u| + C \) for our exercise.
- After applying the constant multiple \(-\frac{1}{4}\) outside the integral, the complete result becomes \(-\frac{1}{4} \ln |u| + C\).
Indefinite Integral
An indefinite integral is an integral that contains a constant of integration and does not have upper and lower limits. It represents a family of functions rather than a single numerical value.
- In the exercise provided, the integral \( \int \frac{dx}{1-4x} \) was solved as an indefinite integral, resulting in \(-\frac{1}{4} \ln |1 - 4x| + C\).
- The \( C \) represents the constant of integration, which accounts for any constant that could be added to a particular solution to make it a general solution.
- This forms a family of functions that are all antiderivatives of the original function.
