Problem 15
Question
Find each indefinite integral. \(\int \frac{3 d x}{x}\)
Step-by-Step Solution
Verified Answer
The indefinite integral is \( 3 \ln |x| + C \).
1Step 1: Understand the Problem
We are asked to find the indefinite integral of \( \int \frac{3}{x} \, dx \). The goal is to determine the antiderivative, which is a function whose derivative would yield \( \frac{3}{x} \). Indefinite integrals also include a constant \( C \), which represents the family of all possible antiderivatives.
2Step 2: Identify the Integral Formula
Recognize that \( \frac{1}{x} \) is a derivative of \( \ln|x| \). Therefore, \( \int \frac{1}{x} \, dx \) gives \( \ln |x| + C \). We can use this fact in our problem.
3Step 3: Apply the Formula and Solve
The problem involves \( \frac{3}{x} \), which is a constant multiplied by \( \frac{1}{x} \). According to integral rules, \( \int \frac{3}{x} \, dx = 3 \int \frac{1}{x} \, dx \). Use the result from Step 2, we get \( 3(\ln |x| + C) \).
4Step 4: Simplify the Solution
Simplifying the expression from step 3, we obtain \( 3 \ln |x| + C \), where \( C \) remains an arbitrary constant of integration.
Key Concepts
AntiderivativeConstant of IntegrationLogarithmic Integration
Antiderivative
In calculus, finding an antiderivative of a function is akin to running differentiation backward. The antiderivative of a function is another function that differentiates to give the original function. For example, if you have a function f(x) and its antiderivative F(x), then the derivative of F(x) is f(x). When it comes to indefinite integration, we are essentially finding the general form of the antiderivative. This is denoted by the symbol \( \int \). A key point is that each function has infinitely many antiderivatives, differentiated by the constant \( C \), known as the constant of integration.
Understanding antiderivatives is essential as they help in understanding rates of change, a fundamental concept in many scientific fields like physics and engineering.
Understanding antiderivatives is essential as they help in understanding rates of change, a fundamental concept in many scientific fields like physics and engineering.
- An antiderivative is not unique; rather, it is a family of functions.
- Performing differentiation on an antiderivative should provide the original function.
Constant of Integration
When working with indefinite integrals, there is an element always present but often overlooked: the constant of integration. In the process of finding antiderivatives, indefinite integrals erase the precision of exact function values due to the inclusion of an arbitrary constant. This constant, denoted as \( C \), represents all possible values the antiderivative can take.
This is crucial because integrating removes certain information about the starting point. When calculating the indefinite integral \( \int \frac{3}{x} \, dx \), we would usually obtain a result like \( 3 \ln |x| + C \). Here, the \( C \) allows the solution to encompass a whole family of functions.
This is crucial because integrating removes certain information about the starting point. When calculating the indefinite integral \( \int \frac{3}{x} \, dx \), we would usually obtain a result like \( 3 \ln |x| + C \). Here, the \( C \) allows the solution to encompass a whole family of functions.
- The constant of integration, \( C \), is what makes an integral indefinite.
- It signifies all the vertical shifts of the antiderivative on a graph.
- Without \( C \), the solution would refer to one specific antiderivative only.
Logarithmic Integration
Logarithmic integration plays a vital role when dealing with the integration of inverse functions such as \( \frac{1}{x} \). This specific type of integration relies on knowing that the derivative of the natural logarithm function \( \ln|x| \) is \( \frac{1}{x} \). This property is highly useful in solving integral problems that include terms like \( \frac{1}{x} \).
In our original problem, \( \int \frac{3}{x} \, dx \), we use this concept by recognizing that \( \int \frac{1}{x} \, dx = \ln |x| + C \), enabling us to rewrite the integral as a multiple of \( \ln|x| \) with respect to \( \frac{3}{x} \).
In our original problem, \( \int \frac{3}{x} \, dx \), we use this concept by recognizing that \( \int \frac{1}{x} \, dx = \ln |x| + C \), enabling us to rewrite the integral as a multiple of \( \ln|x| \) with respect to \( \frac{3}{x} \).
- Logarithmic integration is foundational for integrating functions of the form \( \frac{1}{x} \).
- It applies the natural logarithm \( \ln|x| \) due to its derivative property.
- This technique simplifies finding antiderivatives in many practical scenarios.
Other exercises in this chapter
Problem 15
Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int e^{-x^{2}} x d x $$
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Use a graphing calculator Riemann Sum program from the Internet (see page 332 ) to find the following Riemann sums. i. Calculate the Riemann sum for each functi
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