Problem 33 - Calculus: Early Transcendentals

Question

Evaluate the integral. \( \displaystyle \int^{2}_{1} \biggl( \frac{x}{2} - \frac{2}{x}\biggr) \,dx \)

Step-by-Step Solution

Answer

\( \frac{3}{4} - 2\ln 2 \) is the evaluated integral.

Step 1: Break Down the Integral Expression

The given integral is \( \int_{1}^{2} \left( \frac{x}{2} - \frac{2}{x} \right) \, dx \). Note that it is a definite integral from 1 to 2. The integrand can be split into two separate integrals: \( \int_{1}^{2} \frac{x}{2} \, dx - \int_{1}^{2} \frac{2}{x} \, dx \).

Step 2: Evaluate the First Integral Component

Let's evaluate the first integral \( \int_{1}^{2} \frac{x}{2} \, dx \). We can factor out the constant \( \frac{1}{2} \) from the integral: \( \frac{1}{2} \int_{1}^{2} x \, dx \). The antiderivative of \( x \) is \( \frac{x^2}{2} \). Thus, we have: \[ \frac{1}{2} \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \left( \frac{4}{2} - \frac{1}{2} \right) = \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}. \]

Step 3: Evaluate the Second Integral Component

Now evaluate the second integral \( \int_{1}^{2} \frac{2}{x} \, dx \). The antiderivative of \( \frac{2}{x} \) is \( 2\ln|x| \). Substituting the limits, we get: \[ \left[ 2\ln|x| \right]_{1}^{2} = 2\ln 2 - 2\ln 1 = 2\ln 2. \] Since \( \ln 1 = 0 \), the expression simplifies to \( 2\ln 2 \).

Step 4: Combine the Results

Subtract the second integral result from the first as per the original expression: \( \frac{3}{4} - 2\ln 2 \). This is the evaluated value of the definite integral.

Key Concepts

Antiderivatives

Antiderivatives, also known as indefinite integrals, are essentially the reverse process of differentiation. Finding the antiderivative means determining a function whose derivative is the original function given in the integral. This operation does not include the limits of integration.

• An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \).
• For example, the antiderivative of \( x \) is \( \frac{x^2}{2} + C \), where \( C \) is a constant representing the family of all antiderivatives.

Knowing how to find antiderivatives helps evaluate definite integrals, which involves finding differences between antiderivative values at given limits. In our previous exercise, \( \int x \, dx \) was computed by first finding its antiderivative, \( \frac{x^2}{2} \). Afterward, it was evaluated at specific boundaries.

Logarithmic Integration

Logarithmic integration is a technique applied when the integrand involves expressions like \( \frac{1}{x} \). When dealing with integrals of this form, the antiderivative is related to natural logarithms.

• The general rule is that \( \int \frac{1}{x} \, dx = \ln|x| + C \).
• This property is used when functions can be expressed in parts involving \( \frac{1}{x} \).

In the problem, \( \int \frac{2}{x} \, dx \) was evaluated using logarithmic integration, resulting in \( 2 \ln|x| \). Understanding this concept allows us to solve integrals with expressions of \( \frac{2}{x} \), yielding a result involving a natural logarithm.

Integration Methods

Integration methods refer to the different techniques used to solve integrals, which can be definite or indefinite. These methods are crucial for evaluating integrals of various forms:

• **Basic Integration:** This involves straightforward integrals that require simple antiderivatives.
• **Substitution and Integration by Parts:** More complex functions might need methods like substitution or integration by parts.
• **Partial Fraction Decomposition:** Useful for breaking down rational functions.
• **Special Functions:** Techniques for integrating logarithmic, exponential, and trigonometric functions.

In the provided exercise, basic integration techniques and understanding of antiderivatives were sufficient. The integral was broken into simpler components, each of which was a typical case where a fixed integration method was applied. This decomposition caters the overall problem and provides straightforward results.

Limits of Integration

In definite integrals, the limits of integration define the interval over which the integration is performed. These limits transform an indefinite integral into a definite one, producing a specific numerical result.

• **Lower and Upper Limits:** Notated as \( \int_{a}^{b} \), where \( a \) is the lower limit and \( b \) is the upper limit.
• **Application:** Calculate the antiderivative, and then evaluate it at both limits of integration, subtracting the lower result from the upper.

In our exercise, the integral \( \int_{1}^{2} \) defines the range from 1 to 2. Each component was evaluated at these bounds, combining antiderivative findings at both points. This concept ensures that the evaluated integral represents the net area under the curve from the specified start to end points.