Problem 35 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(29-50.\) $$\int_{\ln 4}^{\ln 9} e^{x / 2} d x$$

Step-by-Step Solution

Answer

The evaluated integral is 2.

Step 1: Recognize the Integral's Form

The integral given is \( \int_{\ln 4}^{\ln 9} e^{x / 2} \, dx \). This requires evaluating a definite integral of an exponential function of the form \( e^{u} \) where \( u = \frac{x}{2} \).

Step 2: Perform Substitution

To simplify the integral, use the substitution \( u = \frac{x}{2} \), which implies that \( du = \frac{1}{2} dx \) or \( dx = 2 \, du \). Also, adjust the limits of integration: when \( x = \ln 4 \), \( u = \frac{\ln 4}{2} \) and when \( x = \ln 9 \), \( u = \frac{\ln 9}{2} \).

Step 3: Transform the Integral

Substitute \( u \) and \( dx \) in the integral: \[\int_{\ln 4}^{\ln 9} e^{x / 2} \, dx = \int_{\frac{\ln 4}{2}}^{\frac{\ln 9}{2}} e^u \, (2 \, du) = 2 \int_{\frac{\ln 4}{2}}^{\frac{\ln 9}{2}} e^u \, du\].

Step 4: Integrate the Function

The antiderivative of \( e^u \) is simply \( e^u \), so the integral becomes \[2 \left[ e^u \right]_{\frac{\ln 4}{2}}^{\frac{\ln 9}{2}}\].

Step 5: Evaluate the Definite Integral

Apply the limits of integration: \[2 \left( e^{\frac{\ln 9}{2}} - e^{\frac{\ln 4}{2}} \right)\].This simplifies using the property \( e^{\ln a^b} = a^b \) to \[2 \left( (9^{1/2}) - (4^{1/2}) \right) = 2 (3 - 2)\].

Step 6: Simplify the Result

Upon simplification, we get:\[2 \times 1 = 2\].

Key Concepts

Exponential Functions

Exponential functions are one of the most important mathematical concepts, used widely in various fields such as science, engineering, and finance. An exponential function is typically expressed in the form \( f(x) = a \, e^{bx} \), where:

• \( a \) is a constant, known as the coefficient.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.
• \( b \) is a constant that dictates the function's growth or decay rate.

Understanding how to work with exponential functions is crucial. This involves recognizing their form, interpreting their behavior, and analyzing their properties. In the context of calculus and integration, exponential functions often appear in problems related to growth processes, such as population growth or radioactive decay. When integrating an exponential function like \( e^{x/2} \), it's essential to apply methods like substitution to simplify the integral for easier computation.

Substitution Method

The substitution method is a useful technique for solving integrals that are difficult to evaluate in their original form. This method involves replacing the original variable with a new variable, aiming to simplify the integral's structure. Here's how it works:

• Select a new variable, \( u \), which represents a function of the original variable, \( x \).
• Express \( dx \) in terms of \( du \) by differentiating \( u \) with respect to \( x \).
• Substitute \( u \) and \( du \) into the original integral.

In our example, the substitution \( u = \frac{x}{2} \) was chosen, which simplifies the integrand from \( e^{x/2} \) to \( e^u \). The derivative \( du = \frac{1}{2} dx \) is used to express \( dx \) as \( 2 \, du \). Thus, the integral becomes easier to evaluate: \( 2 \int e^u \, du \). The substitution method not only simplifies computation but also aids in recognizing antiderivatives during evaluation.

Antiderivatives

An antiderivative of a function is another function whose derivative gives back the original function. For exponential functions, recognizing antiderivatives is straightforward due to their special properties. If we have a function like \( e^u \), its antiderivative is simply itself: \( e^u \). This makes integration quite direct.
When evaluating integrals, antiderivatives are crucial. They allow us to move from the indefinite integral to the definite integral by finding a function that, when differentiated, returns the original integrand. In our example, the antiderivative of \( e^u \) is used to find the value of the definite integral:
\[2 \left[ e^u \right]_{\frac{\ln 4}{2}}^{\frac{\ln 9}{2}}\]By substituting back the limits, you accurately compute the integral's result.

Integration Limits

Integration limits define the interval over which we calculate the area under a function's curve. They are two values that bound the region of interest for integration. For definite integrals, these limits are crucial in determining the exact value of the area.
For the given exercise, the original limits were \( x = \ln 4 \) and \( x = \ln 9 \). After substitution, these limits transform according to the rule \( u = \frac{x}{2} \), giving us new limits: \( u = \frac{\ln 4}{2} \) and \( u = \frac{\ln 9}{2} \).

• Upper limit at \( x = \ln 9 \) changes to \( u = \frac{\ln 9}{2} \).
• Lower limit at \( x = \ln 4 \) changes to \( u = \frac{\ln 4}{2} \).

These transformed limits ensure the integral is evaluated correctly over the appropriate range, which is paramount in achieving the correct result, as integration with incorrect limits could lead to incorrect conclusions about the problem at hand.