Problem 58
Question
Compute the indefinite integrals. $$ \int 2 e^{3 x} d x $$
Step-by-Step Solution
Verified Answer
\( \frac{2}{3} e^{3x} + C \)
1Step 1: Understand the Integral
The given problem is to compute the indefinite integral of the function \(2 e^{3x}\). An indefinite integral is essentially the antiderivative of a function.
2Step 2: Identify the Integral Formula
Recognize that the integral \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \(a\) is a constant and \(C\) is the constant of integration.
3Step 3: Apply the Integral Formula
For the integral \( \int 2e^{3x} \, dx \), apply the formula with \(a = 3\). This yields \( 2 \times \frac{1}{3} e^{3x} = \frac{2}{3} e^{3x} + C \).
4Step 4: Simplify the Expression
The expression simplifies to \( \frac{2}{3} e^{3x} + C \) where \(C\) is the constant of integration.
Key Concepts
Understanding the AntiderivativeApplying the Integral FormulaExploring the Constant of Integration
Understanding the Antiderivative
In calculus, finding the antiderivative of a function is synonymous with performing the operation of an indefinite integral. The antiderivative tells us what function we originally derived to get our current function. It's like working backward.
When you're asked to integrate a function like \( 2e^{3x} \), you're essentially finding its antiderivative. This process involves determining a function whose derivative is the given function.
For instance, if you take the derivative of \( \frac{2}{3}e^{3x} \), you end up with \( 2e^{3x} \). Hence, \( \frac{2}{3}e^{3x} \) is the antiderivative of \( 2e^{3x} \). In this case, the antiderivative helps us find the original function before differentiation.
When you're asked to integrate a function like \( 2e^{3x} \), you're essentially finding its antiderivative. This process involves determining a function whose derivative is the given function.
For instance, if you take the derivative of \( \frac{2}{3}e^{3x} \), you end up with \( 2e^{3x} \). Hence, \( \frac{2}{3}e^{3x} \) is the antiderivative of \( 2e^{3x} \). In this case, the antiderivative helps us find the original function before differentiation.
Applying the Integral Formula
The integral formula is a key tool in finding antiderivatives. For exponential functions like \( e^{ax} \), there is a specific formula:
When integrating \( 2e^{3x} \), the constant \( 2 \) should be multiplied by the integral of \( e^{3x} \). So, by substituting \( a = 3 \) into the formula, we get \( \frac{1}{3}e^{3x} \). Once we multiply by \( 2 \), it results in \( \frac{2}{3}e^{3x} \).
The integral formula provides a straightforward method to handle derivatives of exponential functions easily.
- \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \)
When integrating \( 2e^{3x} \), the constant \( 2 \) should be multiplied by the integral of \( e^{3x} \). So, by substituting \( a = 3 \) into the formula, we get \( \frac{1}{3}e^{3x} \). Once we multiply by \( 2 \), it results in \( \frac{2}{3}e^{3x} \).
The integral formula provides a straightforward method to handle derivatives of exponential functions easily.
Exploring the Constant of Integration
The concept of the constant of integration, represented as \( C \), is crucial when performing indefinite integrals.
Why do we need it? It accounts for all potential original functions that could have led to the derivative in question. Since the derivative of a constant is always zero, any constant can be added to the antiderivative without changing the derivative.
In our exercise, the result of integrating \( 2e^{3x} \) therefore includes \( + C \) to signify all such possible constants that satisfy the original integral.
Why do we need it? It accounts for all potential original functions that could have led to the derivative in question. Since the derivative of a constant is always zero, any constant can be added to the antiderivative without changing the derivative.
- The constant \( C \) represents any number that could have been lost when the derivative was first taken.
- Without \( C \), the indefinite integral wouldn't accurately represent all possible solutions.
In our exercise, the result of integrating \( 2e^{3x} \) therefore includes \( + C \) to signify all such possible constants that satisfy the original integral.
Other exercises in this chapter
Problem 57
Given that \(\int_{0}^{a} x^{3} d x=\frac{1}{4} a^{4}\), evaluate the following integrals: (a) \(\int_{0}^{2} x^{3} d x\) (b) \(\int_{0}^{1} 2 x^{3} d x\) (c) \
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Given that \(\int_{0}^{a} x^{4} d x=\frac{1}{5} a^{5}\) evaluate the following integrals (a) \(\int_{0}^{2} x^{4} d x\) (b) \(\int_{0}^{1} \frac{x^{4}}{2} d x\)
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Compute the indefinite integrals. $$ \int 3 e^{-x} d x $$
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