Problem 39 - Calculus Single Variable

Question

Evaluate the given definite integral by finding an antiderivative of the integrand and applying Theorem \(3 .\) $$ \int_{0}^{3} 2^{x} d x $$

Step-by-Step Solution

Answer

The evaluated definite integral is \( \frac{7}{\ln(2)} \).

Step 1: Identify the Integrand

The integrand in the definite integral is \(2^x\). We need to find an antiderivative for this function.

Step 2: Find the General Antiderivative

The antiderivative of \(2^x\) involves exponential functions. Generally, the antiderivative of \(a^x\) is \(\frac{a^x}{\ln(a)}\). Thus, the antiderivative of \(2^x\) is \(\frac{2^x}{\ln(2)}+C\), where \(C\) is the constant of integration.

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that to evaluate a definite integral \(\int_{a}^{b} f(x) \, dx\), find the antiderivative \(F(x)\), and compute \(F(b) - F(a)\). Here, \(F(x) = \frac{2^x}{\ln(2)}\) is the antiderivative.

Step 4: Evaluate the Antiderivative at the Bounds

Plug the upper bound and lower bound into the antiderivative and subtract: \[ F(3) - F(0) = \frac{2^3}{\ln(2)} - \frac{2^0}{\ln(2)} = \frac{8}{\ln(2)} - \frac{1}{\ln(2)}. \]

Step 5: Simplify the Expression

Combine the terms: \[ \frac{8}{\ln(2)} - \frac{1}{\ln(2)} = \frac{8 - 1}{\ln(2)} = \frac{7}{\ln(2)}. \] This is the value of the definite integral.

Key Concepts

Antiderivative

Understanding the concept of an antiderivative is crucial when solving problems involving definite integrals. An antiderivative is a function whose derivative is the original function you're dealing with.
The process of finding an antiderivative is often referred to as integration, the reverse operation of differentiation.
For instance, given a function like \(2^x\), its antiderivative is \(\frac{2^x}{\ln(2)} + C\). This result follows a general rule for exponential functions \(a^x\), where the antiderivative is \(\frac{a^x}{\ln(a)} + C\).

• \(C\) represents the constant of integration, which is essential for the general solution of indefinite integrals.
• When solving definite integrals, this constant cancels out, so you often don’t see it in the final answer.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the processes of differentiation and integration, showing how they are reverse operations. It consists of two parts, but for definite integrals, we focus on one main concept.
When evaluating a definite integral, this theorem states that if you find an antiderivative \(F(x)\) of \(f(x)\), you can evaluate the definite integral \(\int_{a}^{b} f(x) \, dx\) as \(F(b) - F(a)\).

• This method simplifies the calculation by reducing it to simple arithmetic once the antiderivative is found.
• In our example, this means calculating \(F(3) - F(0)\) for the antiderivative \(F(x) = \frac{2^x}{\ln(2)}\).

The beauty of this method is its efficiency, transforming complex integration problems into manageable tasks.

Exponential Functions

Exponential functions are central in many math problems, including integration exercises. These functions, typically in the form of \(a^x\), where \(a\) is a constant, exhibit rapid growth or decay.
They have unique integration rules because their derivatives are tied closely to their original forms. For instance, the derivative of \(e^x\) is \(e^x\), indicative of their special nature.

• For a base other than \(e\), such as \(2\), the derivative becomes \(2^x \ln(2)\), necessitating the reciprocal adjustment when integrating.
• This reciprocal relationship is why the antiderivative of \(2^x\) involves \(\ln(2)\) in the denominator.

These properties enable us to handle a broad range of real-world problems, from growth models to signals processing, making understanding exponential functions invaluable.