Problem 17 - Applied Calculus

Question

Evaluate the integrals in both exactly [e.g. \(\ln (3 \pi)] \text { and numerically [e.g. } \ln (3 \pi) \approx 2.243]\). $$\int_{1}^{5} \ln t d t$$

Step-by-Step Solution

Answer

Exact: \( 5 \ln 5 - 4 \); Numerical: \( \approx 4.047 \).

Step 1: Identify the Function to be Integrated

The given integral is \( \int_{1}^{5} \ln t \, dt \). The function we need to integrate is \( \ln t \).

Step 2: Choose an Integration Method

For the function \( \ln t \), use integration by parts, a method where \( \int u \, dv = uv - \int v \, du \). Choose \( u = \ln t \) and \( dv = dt \).

Step 3: Differentiate and Integrate Components

Differentiate \( u = \ln t \) to get \( du = \frac{1}{t} dt \). Integrate \( dv = dt \) to obtain \( v = t \).

Step 4: Apply Integration by Parts Formula

Substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula: \[ \int \ln t \, dt = t \ln t - \int t \cdot \frac{1}{t} dt = t \ln t - \int dt. \]

Step 5: Solve the Remaining Integral

The remaining integral is \( \int dt = t \). Thus, substituting back gives \( t \ln t - t + C \).

Step 6: Evaluate the Definite Integral

Apply the limits from 1 to 5: \[ \left[ t \ln t - t \right]_{1}^{5} = \left( 5 \ln 5 - 5 \right) - \left( 1 \cdot \ln 1 - 1 \right). \]

Step 7: Simplify

Note that \( \ln 1 = 0 \). Simplifying gives: \( 5 \ln 5 - 5 - (0 - 1) = 5 \ln 5 - 4 \).

Step 8: Numerical Approximation

Calculate the numerical value: \( 5 \ln 5 - 4 \approx 4.047 \) using a calculator.

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation. It's particularly useful when faced with integrals of products of functions, especially when one of the functions is easy to differentiate while the other is easy to integrate. In the formula \( \int u \, dv = uv - \int v \, du \), \( u \) and \( dv \) are chosen strategically:

• **Choose \( u \):** For this problem, \( u = \ln t \) is chosen because its derivative, \( du = \frac{1}{t} dt \), simplifies the integration process.
• **Choose \( dv \):** Naturally, the remaining part of the integral \( dv = dt \) is selected, which integrates to \( v = t \).

Once \( u \) and \( dv \) are identified, apply the integration by parts formula. Substitute all parts, perform the integration, and simplify. Always remember to substitute back into the original integral's limits if dealing with definite integrals.
Integration by parts is not always straightforward and can sometimes require additional algebraic manipulation. However, with practice, it becomes a powerful tool in evaluating complex integrals.

Natural Logarithms

Natural logarithms, denoted as \( \ln \), are logarithms with base \( e \), where \( e \) is an irrational constant approximately equal to 2.718.

Natural logarithms have several unique properties that make them very useful:

• **Logarithm of 1 is Zero:** \( \ln 1 = 0 \). This key property simplifies calculations, as seen when evaluating definite integrals.
• **Derivative of Natural Logarithm:** The derivative of \( \ln x \) is \( \frac{1}{x} \). This property is instrumental during integration by parts, allowing simplification and transformation of integrals.
• **Product and Quotient Rules:** For sums or differences, natural logarithms allow expansion using the rules \( \ln(ab) = \ln a + \ln b \) and \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).

These properties make the natural log essential in calculus and real-world applications. When handling problems like the integral of \( \ln t \), knowing how it behaves under integration and its numerical evaluation of bounds can greatly simplify the work.

Numerical Approximation

Numerical approximation is the process of finding approximate numerical values for expressions or constants that cannot be easily calculated exactly. It can be particularly helpful in calculus when dealing with integrals that do not have simple antiderivatives or require computational assistance.

In our context of computing \( \int_{1}^{5} \ln t \, dt \), we initially find the exact result, but obtaining a numerical approximation, such as \( 5 \ln 5 - 4 \approx 4.047 \), helps in understanding the magnitude and behavior of the integral.

Here are ways numerical approximation plays a role:

• **Calculator Usage:** Modern calculators or software can evaluate expressions like \( 5 \ln 5 \) to a decimal approximation at the press of a button.
• **Error Consideration:** Numerical methods have inherent errors, hence approximate values like \( 4.047 \) are sufficiently close for practical purposes but can differ slightly due to computational precision.
• **Practicality:** In real-world applications, exact values might not be necessary, and a numerical approximation can provide timely insights or decisions.

Understanding numerical approximation is crucial, as it bridges exact mathematical results with practical, real-world computational needs.