Problem 82
Question
Compute the indefinite integrals. $$ \int \frac{2 x+5}{x} d x $$
Step-by-Step Solution
Verified Answer
The integral of \( \int \frac{2x+5}{x} \, dx \) is \( 2x + 5\ln|x| + C \).
1Step 1: Simplify the integrand
First, we need to simplify the integrand \( \frac{2x + 5}{x} \). This can be done by splitting the fraction into separate terms: \( \frac{2x}{x} + \frac{5}{x} \). This simplifies to \( 2 + \frac{5}{x} \).
2Step 2: Integrate each term separately
The integral of \( 2 \) with respect to \( x \) is straightforward: \( \int 2 \, dx = 2x \). For \( \frac{5}{x} \), we use the rule that \( \int \frac{1}{x} \, dx = \ln|x| \). Hence, \( \int \frac{5}{x} \, dx = 5 \ln|x| \).
3Step 3: Combine the integrals and add the constant of integration
Combine the results from the previous step to form the complete integral. We have \( 2x + 5 \ln|x| \). Don't forget to add the constant of integration \( C \), since it's an indefinite integral. So the final answer is \( 2x + 5 \ln|x| + C \).
Key Concepts
Understanding IntegrationThe Constant of IntegrationIntegrand Simplification Techniques
Understanding Integration
Integration is a fundamental concept in calculus, often considered the reverse process of differentiation. It allows us to determine the accumulated value of a function over an interval, which can be visualized as finding the area under the curve described by the function. When we perform integration without specific limits, we are dealing with indefinite integrals.
An indefinite integral, expressed as \( \int f(x) \, dx \), represents the collection of all antiderivatives of the function \( f(x) \). Since antiderivatives are not unique (they differ by a constant), indefinite integrals always include a constant of integration \( C \).
To solve an integration problem, we usually identify the function to integrate and apply integration rules or formulas, such as the power rule or properties of logarithmic functions. Remember:
An indefinite integral, expressed as \( \int f(x) \, dx \), represents the collection of all antiderivatives of the function \( f(x) \). Since antiderivatives are not unique (they differ by a constant), indefinite integrals always include a constant of integration \( C \).
To solve an integration problem, we usually identify the function to integrate and apply integration rules or formulas, such as the power rule or properties of logarithmic functions. Remember:
- The integral symbol \( \int \) signifies the operation of integration.
- \( dx \) indicates the variable of integration, hinting that you are integrating with respect to \( x \).
The Constant of Integration
When dealing with indefinite integrals, the constant of integration is an essential component. The primary reason behind using this constant, denoted as \( C \), is that integration generally has multiple antiderivatives. This occurs because taking the derivative of any constant equals zero.
For instance, the integral \( \int x^2 \, dx \) is \( \frac{x^3}{3} + C \). The \( C \) represents any constant value that could have been part of the original function before differentiation. Since we cannot determine this constant when dealing with indefinite integrals, it is important to include \( C \) in the final solution.
Why is it significant? Consider the equation of motion derived from physics. The constant of integration could portray an initial velocity or position. Thus, ensuring we always add \( C \) provides completeness and accuracy in mathematical and scientific analyses.
For instance, the integral \( \int x^2 \, dx \) is \( \frac{x^3}{3} + C \). The \( C \) represents any constant value that could have been part of the original function before differentiation. Since we cannot determine this constant when dealing with indefinite integrals, it is important to include \( C \) in the final solution.
Why is it significant? Consider the equation of motion derived from physics. The constant of integration could portray an initial velocity or position. Thus, ensuring we always add \( C \) provides completeness and accuracy in mathematical and scientific analyses.
Integrand Simplification Techniques
One of the first steps in solving an integration problem can involve simplifying the integrand, which is the function under the integral sign. Simplifying the integrand can make the integration process much more manageable.
In our exercise, the original integrand was \( \frac{2x + 5}{x} \). To simplify it, you can break it down by separating the numerator and distributing the fraction: \( \frac{2x}{x} + \frac{5}{x} \). This makes it clear that the integrand simplifies to \( 2 + \frac{5}{x} \), which is a much easier expression to integrate term by term.
Simplifying offers several advantages:
In our exercise, the original integrand was \( \frac{2x + 5}{x} \). To simplify it, you can break it down by separating the numerator and distributing the fraction: \( \frac{2x}{x} + \frac{5}{x} \). This makes it clear that the integrand simplifies to \( 2 + \frac{5}{x} \), which is a much easier expression to integrate term by term.
Simplifying offers several advantages:
- Reduces complexity, leading to fewer calculation errors.
- Makes it easier to identify familiar integration formulas.
- Leads to faster integration, especially useful in solving complex problems.
Other exercises in this chapter
Problem 80
Compute the indefinite integrals. $$ \int \frac{1}{x-3} d x $$
View solution Problem 81
Compute the indefinite integrals. $$ \int \frac{2 x-1}{3 x} d x $$
View solution Problem 83
Compute the indefinite integrals. $$ \int \frac{1}{2 x+1} d x $$
View solution Problem 84
Compute the indefinite integrals. $$ \int \frac{1}{3 x-3} d x $$
View solution