Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 59
Evaluate each definite integral. $$ \int_{0}^{1} 12 e^{3 x} d x $$
Short Answer
Expert verified
The value of the definite integral is \(4e^3 - 4\).
Step by step solution
01
Identify the Integral Type
The given integral is a definite integral with respect to the variable \(x\). We need to evaluate \( \int_{0}^{1} 12 e^{3x} \, dx \) over the interval from 0 to 1.
02
Determine the Antiderivative
The integral involves an exponential function of the form \(e^{3x}\). To find the antiderivative, recall the general rule: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). For this problem, \(a = 3\). Therefore, the antiderivative of \(12 e^{3x}\) is \( \frac{12}{3} e^{3x} = 4e^{3x} \).
03
Apply the Fundamental Theorem of Calculus
To evaluate the definite integral, apply the Fundamental Theorem of Calculus, which states that \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \), where \(F(x)\) is an antiderivative of \(f(x)\). Substitute the antiderivative we found: \(4e^{3x}\). So we want \([4e^{3x}]_{0}^{1} = 4e^{3 \times 1} - 4e^{3 \times 0} \).
04
Evaluate the Antiderivative at the Bounds
Compute \(4e^{3 \times 1} = 4e^3\) and \(4e^{3 \times 0} = 4e^0 = 4\). Substitute these into the expression: \(4e^3 - 4\).
05
Simplify the Expression
Finally, simplify the result: \(4e^3 - 4\). This is the evaluated definite integral.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Function
An exponential function is a type of mathematical expression where a constant base is raised to a variable exponent. A common example is the function of the natural base, denoted as \(e\), which is approximately equal to 2.718. The general form of an exponential function is \(y = e^{ax}\), where \(a\) is a constant and \(x\) is the variable.
This type of function is crucial in various fields:
This type of function is crucial in various fields:
- It models growth processes such as populations or radioactive decay.
- Exponential functions are used in continuous interest calculations in finance.
- They are extensively utilized in calculus, especially in solving differential equations.
Antiderivative
The antiderivative, also known as an indefinite integral, is the reverse process of differentiation. It involves finding a function whose derivative is the given function. For instance, if the derivative of \(F(x)\) is \(f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\).
In the context of exponential functions, the process becomes straightforward. Due to the properties of the natural exponent, the derivative of \(e^{ax}\) with respect to \(x\) is \(ae^{ax}\). Consequently, the antiderivative of \(e^{ax}\) can be readily determined as:
In the context of exponential functions, the process becomes straightforward. Due to the properties of the natural exponent, the derivative of \(e^{ax}\) with respect to \(x\) is \(ae^{ax}\). Consequently, the antiderivative of \(e^{ax}\) can be readily determined as:
- \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\)
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It provides a powerful technique to evaluate definite integrals, linking them to antiderivatives. This theorem is composed of two parts:
In practice, this means that if you know an antiderivative of a function, you can calculate the definite integral by substituting the bounds into the antiderivative and subtracting. This forms the basis for many calculus applications, emphasizing why it is such a crucial theorem.
- The first part states that if \(f\) is continuous over an interval \([a, b]\) and \(F\) is an antiderivative of \(f\), then the integral of \(f\) from \(a\) to \(b\) is given by \(F(b) - F(a)\).
- The second part asserts that if \(F\) is a function defined by an integral of a continuous function \(f\), then \(F'(x) = f(x)\).
In practice, this means that if you know an antiderivative of a function, you can calculate the definite integral by substituting the bounds into the antiderivative and subtracting. This forms the basis for many calculus applications, emphasizing why it is such a crucial theorem.
Integration Techniques
Integration techniques refer to methods used to solve integrals, both definite and indefinite. Many functions require specific approaches to find their integrals efficiently. Some critical techniques include:
In the given example, finding the antiderivative of an exponential function with a linear exponent involves recognizing a pattern and applying basic rules without resorting to more complex methods. Proficiency in these techniques is essential for tackling a wide variety of calculus problems.
- Substitution: Useful when an integral contains a composite function, simplifying it through a change of variables.
- Integration by parts: Derived from the product rule of differentiation, ideal for integrals involving products of functions.
- Partial fraction decomposition: Appear when the integrand is a rational function that can be broken down into simpler fractions.
In the given example, finding the antiderivative of an exponential function with a linear exponent involves recognizing a pattern and applying basic rules without resorting to more complex methods. Proficiency in these techniques is essential for tackling a wide variety of calculus problems.
