Problem 43 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-62\). $$ \int_{\ln 2}^{\ln 3} e^{x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to 1.

Step 1: Identify the Antiderivative

The function we need to integrate is \( e^x \). The antiderivative of \( e^x \) with respect to \( x \) is \( e^x \).

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, the definite integral of \( f(x) \) from \( a \) to \( b \) can be found using the antiderivative \( F(x) \) as follows: \[ \int_a^b f(x) \, dx = F(b) - F(a) \]. Here, \( F(x) = e^x \).

Step 3: Substitute the Limits of Integration

Substitute \( x = \ln 3 \) and \( x = \ln 2 \) into the antiderivative: \( F(\ln 3) - F(\ln 2) = e^{\ln 3} - e^{\ln 2} \).

Step 4: Simplify the Expression

Use the property of exponentials that \( e^{\ln a} = a \). Thus, \( e^{\ln 3} = 3 \) and \( e^{\ln 2} = 2 \). Therefore, the expression simplifies to: \( 3 - 2 = 1 \).

Step 5: State the Final Answer

The value of the integral \( \int_{\ln 2}^{\ln 3} e^x \, dx \) is 1.

Key Concepts

Antiderivative

An antiderivative is essentially the opposite of taking a derivative. If you think about how derivatives give us the rate of change or the slope of a function, antiderivatives take us back to the original function that was behind this rate of change. In the exercise, the function given is the exponential function \( e^x \). The beautiful thing about the exponential function \( e^x \) is that its derivative is itself - which means the same applies when finding the antiderivative. So, when you're asked to find the antiderivative of \( e^x \), you're essentially keeping it as is, because the rate of change of \( e^x \) does not affect its form. This self-similar property is what makes the integration of \( e^x \) straightforward.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differentiation and integration, two core concepts in calculus. It tells us that if you take the antiderivative (also known as the indefinite integral) of a function, you can find the area under the curve of that function between two points. This is extremely useful when dealing with definite integrals. The theorem states that if \( F(x) \) is the antiderivative of \( f(x) \), then:

• \( \int_a^b f(x) \, dx = F(b) - F(a) \).

This means to evaluate the integral, you simply take the antiderivative and calculate its value at the upper limit of integration, subtracting its value at the lower limit. In our exercise, the limits \( \ln 3 \) and \( \ln 2 \) help define the start and end of our interval. This method simplifies the computation of the area under the curve of \( e^x \) between these two bounds.

Exponential Functions

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. In calculus, perhaps the most important exponential function is \( e^x \) due to its unique property that the function itself is equal to its own derivative and antiderivative. This property arises from the natural number \( e \), which is approximately equal to 2.71828, and plays a key role in continuous growth processes.

• For example, \( e^{\ln a} = a \) due to the relationship between exponentials and logarithms.
• This property was used in our exercise to simplify \( e^{\ln 3} \) directly to 3 and \( e^{\ln 2} \) directly to 2.

This simplification helps drastically in calculating the results of definite integrals. The exponential function \( e^x \) often pops up in natural and financial sciences due to its continuous growth property, and is integral to understanding growth rates in various contexts.