Problem 99 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(93-106.\) $$\int_{1}^{4} \frac{\ln 2 \log _{2} x}{x} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( 2 (\ln 2)^2 \).

Step 1: Convert the Logarithm

The given integral is \( \int_{1}^{4} \frac{\ln 2 \log_{2} x}{x} \, dx \). First, let's convert the logarithm to natural logarithms using the change of base formula. Recall that, \( \log_{2} x = \frac{\ln x}{\ln 2} \). Substitute this into the integral to get: \[\int_{1}^{4} \frac{\ln 2 \cdot \frac{\ln x}{\ln 2}}{x} \, dx = \int_{1}^{4} \frac{\ln x}{x} \, dx.\] Here, we find that the integral simplifies as the \( \ln 2 \) terms cancel out.

Step 2: Recognize the Integral Form

The integral \( \int \frac{\ln x}{x} \, dx \) is recognized as a standard integral of the natural logarithm. In particular, this is the integrand of \( \int \ln x \, d(\ln x) \), which is a straightforward integral that evaluates to \[ \int \ln x \, d(\ln x) = \frac{(\ln x)^2}{2} + C. \] Here, we have used a basic integral result.

Step 3: Evaluate the Indefinite Integral

Thus, the indefinite integral is \[ \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C. \] You can double-check this by differentiating the result to verify it matches the original integrand. This confirms the correctness of the integration.

Step 4: Evaluate the Definite Integral

Now, use the evaluation formula for definite integrals. Substitute the boundaries of the integral, from the indefinite result. Evaluate from 1 to 4:\[ \frac{(\ln 4)^2}{2} - \frac{(\ln 1)^2}{2}.\] Recall that \( \ln 1 = 0 \), therefore:\[(\ln 1)^2/2 = 0.\] Substituting, \[ \frac{(\ln 4)^2}{2} - 0 = \frac{(\ln 4)^2}{2}.\]

Step 5: Simplification of Results

We know that \( \ln 4 = 2 \ln 2 \) (as \( 4 = 2^2 \)), hence \[ \frac{(\ln 4)^2}{2} = \frac{(2 \ln 2)^2}{2} = \frac{4 (\ln 2)^2}{2} = 2 (\ln 2)^2. \] Therefore, the value of the definite integral is \( 2 (\ln 2)^2 \).

Key Concepts

Change of Base Formula

When dealing with logarithms, you might encounter bases other than the natural base, which is denoted by \( e \). In such cases, the change of base formula is very helpful. This formula allows you to convert a logarithm with any base to a natural logarithm or any other base of your choice. For a logarithm of the form \( \log_{b}(x) \), the change of base formula is given by

• \( \log_{b}(x) = \frac{\ln(x)}{\ln(b)} \)

By applying this formula, you can convert any logarithm to a natural logarithm (\( \ln \)), making integration and differentiation much simpler. In our integral evaluation, \( \log_{2}(x) \) becomes \( \frac{\ln(x)}{\ln(2)} \). This substitution simplifies the integral, as is demonstrated in the step-by-step solution provided.

Natural Logarithms

The natural logarithm, denoted as \( \ln \), uses the base \( e \), which is an irrational constant approximately equal to 2.71828. It is extensively used in calculus because it simplifies differentiation and integration.

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \), making it a natural choice for integration purposes.
• The integral of \( \frac{\ln(x)}{x} \) equals \( \frac{(\ln(x))^2}{2} + C \), which is a result often applied in calculus exercises.

Understanding natural logarithms is vital for efficiently working with integrals and derivatives, particularly when converting logarithms of other bases into a natural log to simplify calculations, as was necessary in our example problem.

Standard Integrals

Standard integrals are essential building blocks in calculus, allowing us to solve more complex integral problems by applying known results. For instance, some standard integral forms include:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \)
• \( \int \frac{1}{x} \, dx = \ln|x| + C \)
• \( \int e^x \, dx = e^x + C \)

These functions and their integrals are frequently encountered in calculations. The exercise used the standard form \( \int \frac{\ln(x)}{x} \, dx \), which directly leads to \( \frac{(\ln(x))^2}{2} + C \) upon integration. Having a solid grasp of standard integrals enables you to evaluate integrals more effectively by connecting them to familiar patterns, as illustrated in our solution steps.

Definite Integral Evaluation

Definite integral evaluation takes the indefinite integral result and applies specific boundaries to find a numerical value over an interval. The general method involves:

• First, compute the indefinite integral of the function.
• Then, apply the limits: \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \).

In the given exercise, after computing the indefinite integral of \( \frac{\ln(x)}{x} \), the expression \( \frac{(\ln(x))^2}{2} \) was evaluated at the boundaries 1 and 4. The calculation simplifies further when \( \ln(1) = 0 \), leading the definite integral to be \( 2 (\ln 2)^2 \). Understanding how to properly evaluate definite integrals by substituting the upper and lower limit values is crucial for obtaining the exact solution to many calculus problems.