Problem 24
Question
In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{3}{e^{-x}} $$
Step-by-Step Solution
Verified Answer
The general antiderivative of \( f(x) = \frac{3}{e^{-x}} \) is \( 3e^{x} + C \).
1Step 1: Rewrite the Given Function
The function is given as \( f(x) = \frac{3}{e^{-x}} \). The first step is to rewrite this function by recognizing that \( \frac{1}{e^{-x}} = e^{x} \). Therefore, the function can be rewritten as \( f(x) = 3e^{x} \).
2Step 2: Recall the Antiderivative Rule for Exponential Functions
The next step is to recall the antiderivative rule for exponential functions. The antiderivative of \( e^{x} \) with respect to \( x \) is \( e^{x} + C \), where \( C \) is the constant of integration.
3Step 3: Find the General Antiderivative
Using the rule from the previous step, the antiderivative of \( 3e^{x} \) is \( 3 \int e^{x} \, dx = 3e^{x} + C \), where \( C \) is the constant of integration.
Key Concepts
Exponential FunctionsConstant of IntegrationIntegration Techniques
Exponential Functions
Exponential functions are a key concept in calculus and mathematics in general. They involve the mathematical expression of a constant raised to a variable power, typically written as \( a^x \), where \( a \) is a constant and \( x \) is the exponent or variable. Exponential growth or decay is often modeled using these functions.
In the context of calculus, understanding exponential functions is crucial when computing derivatives and antiderivatives. The base \( e \), approximately equal to 2.718, is particularly significant. Functions of the form \( e^x \) are prevalent, as they have the unique property that their derivative and antiderivative are the same as the function itself. This makes integrating exponential functions straightforward once you know the basic rules.
When working with an expression like \( f(x) = 3e^x \), the constant multiplier (in this case, 3) can simply be taken outside the integral, simplifying the process of finding the antiderivative.
In the context of calculus, understanding exponential functions is crucial when computing derivatives and antiderivatives. The base \( e \), approximately equal to 2.718, is particularly significant. Functions of the form \( e^x \) are prevalent, as they have the unique property that their derivative and antiderivative are the same as the function itself. This makes integrating exponential functions straightforward once you know the basic rules.
When working with an expression like \( f(x) = 3e^x \), the constant multiplier (in this case, 3) can simply be taken outside the integral, simplifying the process of finding the antiderivative.
Constant of Integration
The constant of integration is a fundamental concept when finding antiderivatives. Whenever we integrate a function to find its antiderivative, we must add an arbitrary constant \( C \). This is because integration is essentially the reverse operation of differentiation, and the derivative of a constant is zero.
The presence of \( C \) in an integral allows us to account for all possible antiderivatives of a function. For example, if the antiderivative of \( f(x) = 3e^x \) is \( 3e^x + C \), then \( C \) ensures that all vertical shifts of the function are included without changing its slope or general shape.
This constant of integration is important in real-world applications where initial conditions or specific values may dictate a particular solution among the infinite possibilities that include different values of \( C \).
The presence of \( C \) in an integral allows us to account for all possible antiderivatives of a function. For example, if the antiderivative of \( f(x) = 3e^x \) is \( 3e^x + C \), then \( C \) ensures that all vertical shifts of the function are included without changing its slope or general shape.
This constant of integration is important in real-world applications where initial conditions or specific values may dictate a particular solution among the infinite possibilities that include different values of \( C \).
- Always include \( C \) when calculating an indefinite integral.
- \( C \) represents the vertical shift in family of solutions.
Integration Techniques
Integration techniques are the strategies used to find antiderivatives of functions. Although some simple functions have straightforward antiderivatives, more complex functions require additional methods.
For exponential functions such as \( e^x \), the integration process is direct due to their unique properties. However, when dealing with products, quotients, or more complex expressions, other techniques may be needed, such as:
For exponential functions such as \( e^x \), the integration process is direct due to their unique properties. However, when dealing with products, quotients, or more complex expressions, other techniques may be needed, such as:
- Substitution: Used when a function contains a composite structure, like \( g(f(x)) \), and involves substituting a part of the integrand.
- Integration by Parts: A method applied to products of functions, based on the product rule for differentiation.
- Partial Fraction Decomposition: Useful for rational functions, breaking them up into simpler fractions which can be integrated individually.
Other exercises in this chapter
Problem 24
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