Problem 19 - Calculus: Early Transcendental Functions

Question

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\ln 2}\left(e^{x / 2}\right)^{2} d x$$

Step-by-Step Solution

Answer

The value of the integral is 1

Step 1: Simplify the Integral

Rewrite the integral by simplifying the expression inside. In this case, \(\left(e^{x / 2}\right)^{2}\) becomes \(e^{x}\). The integral becomes: \(\int_{0}^{\ln 2} e^{x} d x\)

Step 2: Use the Fundamental Theorem of Calculus Part I

Part I of the Fundamental Theorem of Calculus states that if a function is continuous on the interval \([a, b]\) and F is an antiderivative of f on \([a, b]\), then \(\int_{a}^{b} f(x) d x = F(b) - F(a)\). In this case, the antiderivative of \(e^x\) is \(e^x\), so we substitute \(x = 0\) and \(x = \ln(2)\) into \(e^x\). This gives \(e^{\ln(2)} - e^{0}\)

Step 3: Simplify the result

Making use of the facts that \(e^{ln(2)} = 2\) (since \(e\) and \(ln\) are inverse functions) and \(e^0 = 1\), we simplify the expression to get \(2 - 1\)

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with integrals, which are the mathematical representations of accumulating quantities. It essentially involves adding up parts to find a whole, like computing area under a curve. The Fundamental Theorem of Calculus plays a pivotal role here, as it connects differentiation with integration, offering a way to evaluate definite integrals.

• In a definite integral such as \( \int_{a}^{b} f(x) \, dx \), the interval \([a, b]\) is considered, with \( f(x) \) being the function to integrate.
• The result provides the accumulated value of the function from \( a \) to \( b \).
• This process transforms a complex area problem into a simpler evaluation of antiderivatives.

In our case, the calculation process became simpler once the expression \( \left( e^{x/2} \right)^2 \) transformed into \( e^x \). The integral now succinctly represents the area under the curve \( e^x \) from \( x = 0 \) to \( x = \ln 2 \).

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent. Here, the base \( e \) is approximately 2.718, famously known as Euler's number, and an exponential growth or decay involves \( e^{x} \).

• This function exists prominently in natural growth processes and complex economic models.
• For integration, the exponential nature provides a unique trait: the derivative of \( e^{x} \) is itself, \( e^{x} \), making calculations straightforward.

Given the function \( e^{x} \) in the problem, it conveys an exponentially increasing function throughout the domain. This property simplified our solution, as the antiderivative is identical to the original function, creating a seamless calculation process.

Antiderivatives

Antiderivatives, also known as indefinite integrals, represent functions whose derivatives yield the original function. They play an essential role in solving integration problems involving the Fundamental Theorem of Calculus.

• For any function \( f(x) \), an antiderivative \( F(x) \) satisfies \( F'(x) = f(x) \).
• In definite integration using the Fundamental Theorem, \( F(b) - F(a) \) gives the value of \( \int_{a}^{b} f(x) \, dx \).

In our case, \( e^x \) serves as both the function and its antiderivative, simplifying the process. Substituting the upper and lower bounds with \( e^x \), and using \( F(b) - F(a) \), results directly in the answer \( 2 - 1 \), exploiting the fact \( e^{\ln 2} \) resolves neatly into 2.