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Magazine Chapter 1: Problem 191
In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\). $$ \int_{1}^{x} e^{\prime} d t $$
Short Answer
Expert verified
F(x) = e^x - e.
Step by step solution
01
Define the Evaluation Theorem
First, recall the Evaluation or Fundamental Theorem of Calculus which states: If a function \( f \) is continuous on the interval \([a, b]\), and \( F \) is an antiderivative of \( f \) on \([a, b]\), then \[ \int_{a}^{b} f(t) \, dt = F(b) - F(a). \] We will use this theorem to solve the given problem.
02
Identify the Integrand
Look at the integral provided \( \int_{1}^{x} e^{\prime} \, dt \). The integrand here is \( e^{\prime} \), which involves the constant \( e \) raised to a power. Typically, \( e^{\prime} \) isn't a standard function format; check for a typographical error or missing symbol. Often in such contexts, it is helpful to assume the function might be a simple constant or \( e^1 \). Let's assume \( e^{\prime} \) was intended to be \( e^t \).
03
Determine the Antiderivative
Assuming the correct interpretation of the integrand is \( e^t \), identify its antiderivative. The antiderivative of \( e^t \) with respect to \( t \) is \( e^t \), since the derivative of \( e^t \) is itself.
04
Evaluate Using the Limits
Using the antiderivative \( e^t \), apply the Fundamental Theorem of Calculus: \( \int_{1}^{x} e^t \, dt = e^t \bigg|_{1}^{x} = e^x - e^1 \). Here, \( e^1 \) simplifies to \( e \).
05
Express the Solution as a Function
Substitute and simplify to express the integral as a function: \( F(x) = e^x - e \). This function represents the value of the integral evaluated from 1 to \( x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Antiderivative
An antiderivative of a function is essentially its reverse operation from differentiation. In simpler terms, if you're given a function and you find its derivative, the antiderivative takes you back to the original function.
- The concept is central in integral calculus, particularly when using the Fundamental Theorem of Calculus.
- For example, the function for the exponential, such as \(e^t\), is its own antiderivative. So, both differentiation and integration will yield \(e^t\).
- Thus, if you're given an integral of \(e^t\), its antiderivative is indeed \(e^t\) as well.
The Role of Continuous Functions
Continuous functions are absolutely crucial when dealing with integrals, especially when applying the Fundamental Theorem of Calculus.
- A function being continuous on an interval means that it has no breaks, jumps, or holes over that interval.
- This continuity ensures that the integral process, which involves calculating the area under the curve, provides a consistent and meaningful outcome.
- In the context of our problem, we assumed the function \(e^t\), as it is a smooth and continuous function across any interval of interest.
Integral Evaluation Through the Fundamental Theorem
Integral evaluation is the process of calculating the value of a definite integral altogether. This is where the Fundamental Theorem of Calculus steps in beautifully.
- It provides a bridge between differentiation and integration, showing that they are inverse processes.
- In the exercise example, we applied this theorem by finding the antiderivative \(e^t\) and calculating the definite integral by evaluating it at the limits \(1\) to \(x\).
- The final result, \(F(x) = e^x - e\), came from substituting these limits into the antiderivative.
