Problem 12
Question
In Problems 12 through 14, nd antiderivatives for the given functions. In other words, for each function \(f, \quad\) nd a function \(F\) such that \(F^{\prime}=f .\) Check your answers. (a) \(f(x)=e^{3 x}\) (b) \(f(x)=\frac{3}{e^{x}}\)
Step-by-Step Solution
Verified Answer
The antiderivatives of the given functions are \(F(x) = \frac{1}{3}e^{3x} + C\) for \(f(x)=e^{3 x}\) and \(F(x) = -3e^{-x} + C\) for \(f(x)=\frac{3}{e^{x}}\).
1Step 1: Find the antiderivative of \(f(x)=e^{3 x}\)
To find the antiderivative of \(f(x)=e^{3 x}\), apply the rule of integration: \(\int e^{kx} dx = \frac{1}{k}e^{kx} + C\), where \(k\) is any real number and \(C\) stands for the constant of integration. In this case, \(k=3\). So, \(F(x) = \frac{1}{3}e^{3x} + C\).
2Step 2: Check the first answer
Differentiate the resulting function from Step 1 and check whether it gives the original function. The derivative of \(F(x) = \frac{1}{3}e^{3x} + C\) is \(F'(x) = 3*\frac{1}{3}e^{3x} = e^{3x}\), which is the same as the original function as required.
3Step 3: Find the antiderivative of \(f(x)=\frac{3}{e^{x}}\)
To find the antiderivative of \(f(x)=\frac{3}{e^{x}}\), rewrite the function as \(f(x)=3e^{-x}\). Apply now the same rule of integration that we used in Step 1. Now \(k=-1\). So, \(F(x) = -3e^{-x} + C\).
4Step 4: Check the second answer
Differentiate the resulting function from Step 3 and check whether it gives the original function. The derivative of \(F(x) = -3e^{-x} + C\) is \(F'(x) = -(-3)*(e^{-x}) = \frac{3}{e^{x}}\), which is the same as the original function as required.
Key Concepts
Integration RulesConstant of IntegrationExponential Functions Differentiation
Integration Rules
Understanding integration rules is fundamental for solving calculus problems involving antiderivatives. Antiderivatives, also known as indefinite integrals, represent the reverse operation of differentiation. When we integrate a function, we are essentially finding the family of all functions that, when differentiated, give us the original function.
For example, when dealing with exponential functions such as \(e^{kx}\), where \(k\) is a constant, the integration rule states that \(\int e^{kx} dx = \frac{1}{k}e^{kx} + C\). The term \(C\) represents the constant of integration, which we will discuss in more detail in the next section. This rule applies to all exponential functions with a constant multiplied by the variable in the exponent.
In the exercise provided, this rule allows us to find the antiderivatives of \(e^{3x}\) and \(\frac{3}{e^x}\) by treating the constants 3 and -1 appropriately. Remember that integrating a constant multiple of a function means you can factor out the constant and integrate the function normally, then multiply the result by that constant.
For example, when dealing with exponential functions such as \(e^{kx}\), where \(k\) is a constant, the integration rule states that \(\int e^{kx} dx = \frac{1}{k}e^{kx} + C\). The term \(C\) represents the constant of integration, which we will discuss in more detail in the next section. This rule applies to all exponential functions with a constant multiplied by the variable in the exponent.
In the exercise provided, this rule allows us to find the antiderivatives of \(e^{3x}\) and \(\frac{3}{e^x}\) by treating the constants 3 and -1 appropriately. Remember that integrating a constant multiple of a function means you can factor out the constant and integrate the function normally, then multiply the result by that constant.
Constant of Integration
When we speak about the constant of integration, denoted as \(C\), it's crucial to recognize its significance in antiderivatives. This constant represents an infinite number of functions that differ by any constant value, as any constant's derivative is zero.
When integrating a function to find its antiderivative, we always add \(C\) because the process of differentiation would have removed any constant that was originally there. Every antiderivative is, therefore, a family of functions rather than a single function. For the exercise at hand, after finding the antiderivatives \(\frac{1}{3}e^{3x}\) and \(-3e^{-x}\), we included \(+ C\) to acknowledge that any constant could have been in the original function without affecting its derivative.
When integrating a function to find its antiderivative, we always add \(C\) because the process of differentiation would have removed any constant that was originally there. Every antiderivative is, therefore, a family of functions rather than a single function. For the exercise at hand, after finding the antiderivatives \(\frac{1}{3}e^{3x}\) and \(-3e^{-x}\), we included \(+ C\) to acknowledge that any constant could have been in the original function without affecting its derivative.
Exponential Functions Differentiation
Differentiating exponential functions might seem intimidating at first, but it's a straightforward process once you understand the rules. For any exponential function in the form \(ae^{kx}\), where \(a\) and \(k\) are constants, the derivative will be \(ake^{kx}\). This is because the base of the exponential function, \(e\), has a unique property where its derivative is itself.
Let's apply this to our exercise. When differentiating \(\frac{1}{3}e^{3x} + C\), the constant \(C\) disappears (since the derivative of a constant is zero), and using the differentiation rule for exponential functions, we are left with \(e^{3x}\), validating Step 2 of our solution. Similarly, for \(-3e^{-x} + C\), we apply the constant multiple rule in differentiation, which gives us the original function \(\frac{3}{e^{x}}\), confirming our solution in Step 4. This showcases the beauty of calculus: the symmetry between integration and differentiation, especially with exponential functions.
Let's apply this to our exercise. When differentiating \(\frac{1}{3}e^{3x} + C\), the constant \(C\) disappears (since the derivative of a constant is zero), and using the differentiation rule for exponential functions, we are left with \(e^{3x}\), validating Step 2 of our solution. Similarly, for \(-3e^{-x} + C\), we apply the constant multiple rule in differentiation, which gives us the original function \(\frac{3}{e^{x}}\), confirming our solution in Step 4. This showcases the beauty of calculus: the symmetry between integration and differentiation, especially with exponential functions.
Other exercises in this chapter
Problem 12
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