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Magazine Chapter 7: Problem 29
Evaluate the integrals in Exercises \(29-50\) $$\int\left(e^{3 x}+5 e^{-x}\right) d x$$
Short Answer
Expert verified
The integral evaluates to \( \frac{1}{3}e^{3x} - 5e^{-x} + C \).
Step by step solution
01
Identify the Integral Components
The integral given is \( \int (e^{3x} + 5e^{-x}) \, dx \). It is composed of two separate terms: \( e^{3x} \) and \( 5e^{-x} \). We will integrate each term individually and then sum the results.
02
Integrate the First Term
The first term is \( \int e^{3x} \, dx \). The integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \). Here, \( a = 3 \), so the integral of \( e^{3x} \) is \( \frac{1}{3}e^{3x} + C_1 \).
03
Integrate the Second Term
The second term is \( \int 5e^{-x} \, dx \). As before, integrate \( e^{ax} \) using \( \frac{1}{a}e^{ax} \). For \( 5e^{-x} \), \( a = -1 \), so the integral of \( 5e^{-x} \) is \( -5e^{-x} + C_2 \).
04
Combine the Integrated Terms
Add the integrated results from the two terms: \( \frac{1}{3}e^{3x} - 5e^{-x} + C \), where \( C = C_1 + C_2 \) is the constant of integration.
05
Final Answer for the Integral
The integrated result for \( \int (e^{3x} + 5e^{-x}) \, dx \) is \( \frac{1}{3}e^{3x} - 5e^{-x} + C \). We combined all components into a single expression with the constant of integration \( C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
Exponential functions play a crucial role in calculus, especially when dealing with integrals and derivatives. These functions have the form \(f(x) = a^x\) or \(e^{ax}\) in continuous mathematics where \(e\) is Euler's number, approximately equal to 2.71828. They have unique properties that make them particularly easy to work with in calculus.
- Growth or Decay: Exponential functions are often used to model phenomena such as population growth, radioactive decay, and interest calculations due to their rapid increase or decrease.
- Derivative of Exponentials: The derivative of \(e^x\) remains \(e^x\), which is unique and simplifies many calculus operations. For a general \(e^{ax}\), the derivative is \(ae^{ax}\).
- Integration of Exponentials: Integrating exponential functions like \(e^{ax}\) involves reversing the differentiation process, typically resulting in the expression \(\frac{1}{a}e^{ax}\).
Indefinite Integrals
Indefinite integrals, unlike definite integrals, do not have specified limits of integration. Instead, they represent a family of functions whose derivatives are equivalent to the original function. The indefinite integral, sometimes called an antiderivative, is presented with a generic \(+ C\) at the end to denote this family of functions.
- Basic Notation: An indefinite integral is generally written in the form \(\int f(x)dx\), where the integral sign \(\int\) indicates integration of the function \(f(x)\).
- Role in Calculus: Indefinite integrals are necessary for solving problems related to area under curves, solving differential equations, and analyzing continuous growth models.
Constants of Integration
Whenever integrating a function, especially in the context of indefinite integrals, it's vital to include a constant of integration, denoted as \(C\). This constant accounts for the infinite number of antiderivatives that differ by a constant.
- Purpose: The constant of integration represents any constant value that could have been differentiated away, ensuring covering all possible original functions.
- A Common Mistake: Omitting \(C\) in your solutions can lead to incomplete understanding of the solution space and is a frequent error in early calculus practice.
