Problem 66 - Applied Calculus

Question

Evaluate each definite integral. $$ \int_{1}^{2} \frac{(x+1)^{2}}{x^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{3}{2} + 2 \ln(2) \).

Step 1: Simplify the Integrand

Rewrite the expression inside the integral by expanding the numerator before integrating. The original integral is: \( \int_{1}^{2} \frac{(x+1)^2}{x^2} \, dx \).Expanding \( (x + 1)^2 \), we get \( x^2 + 2x + 1 \).Thus, the integrand becomes:\[\int_{1}^{2} \left(\frac{x^2}{x^2} + \frac{2x}{x^2} + \frac{1}{x^2}\right) \, dx = \int_{1}^{2} \left(1 + \frac{2}{x} + \frac{1}{x^2}\right) \, dx\]

Step 2: Integrate Term-by-Term

Now integrate each term separately over the interval \([1, 2]\):\[\int_{1}^{2} \left( 1 \right) \, dx + \int_{1}^{2} \left( \frac{2}{x} \right) \, dx + \int_{1}^{2} \left( \frac{1}{x^2} \right) \, dx\]- \( \int_{1}^{2} 1 \, dx = [x]_{1}^{2} \)- \( \int_{1}^{2} \frac{2}{x} \, dx = 2[\ln|x|]_{1}^{2} \)- \( \int_{1}^{2} \frac{1}{x^2} \, dx = [-\frac{1}{x}]_{1}^{2} \)

Step 3: Evaluate the Integrals

Calculate the definite integrals obtained in Step 2:- For \( [x]_{1}^{2} \), calculate: \( 2 - 1 = 1 \)- For \( 2[\ln|x|]_{1}^{2} \), calculate: \( 2[\ln(2) - \ln(1)] = 2 \ln(2) \)- For \( [-\frac{1}{x}]_{1}^{2} \), calculate: \(-\frac{1}{2} - (-1) = \frac{1}{2} \)Add the results: \( 1 + 2 \ln(2) + \frac{1}{2} \)

Step 4: Sum the Results

Combine all the evaluated parts:\[1 + 2 \ln(2) + \frac{1}{2} = \frac{3}{2} + 2 \ln(2)\]

Step 5: Final Simplified Answer

Thus, the value of the definite integral is \( \frac{3}{2} + 2 \ln(2) \).

Key Concepts

Integration by Parts

Integration by parts is a technique used in calculus to integrate products of functions. It helps when the standard integration methods don’t work straightforwardly. You use the formula:
\[ \int u \, dv = uv - \int v \, du \]
- **Identify the parts:** Often, one function gets differentiated (typically a polynomial) and the other integrated (commonly exponential or trigonometric).- **Calculate components:** Choose which function in your integral should be "u" and which should be "dv."- **Substitution:** Substitute into the formula to simplify and solve the integral.
This method is particularly useful when dealing with products such as polynomial multiplied by a logarithmic or exponential function, though in this problem it wasn't needed since simplification was more appropriate.

Polynomial Expansion

Polynomial expansion simplifies expressions with powers. It's crucial for converting a product of sums into a sum of products.
In the exercise, \( (x + 1)^2 \) is expanded to \( x^2 + 2x + 1 \). This transforms the integrand into terms that are easier to manipulate:- \( \int \frac{x^2}{x^2} \) becomes \( \int 1 \).- \( \int \frac{2x}{x^2} \) becomes \( \int \frac{2}{x} \).- \( \int \frac{1}{x^2} \) remains \( \int \frac{1}{x^2} \).Breaking it down this way simplifies the integration process. Each term fits a well-known integration formula, making computation straightforward. It's a powerful technique because it handles complex expressions by reducing them to simpler, manageable terms.

Logarithmic Integration

Logarithmic integration refers to integrating functions like \( \frac{1}{x} \), where the output involves logarithms.
For the term \( \int \frac{2}{x} \, dx \), the antiderivative is \( 2 \ln|x| \).Some essential points about logarithmic integration are:

• The integral of \( \frac{1}{x} \) is \( \ln|x| \), and the constant factor "2" remains outside the integral.
• Always consider the absolute value inside the logarithm to ensure it's defined across all real numbers where the function exists.

Evaluating from 1 to 2 gives terms evaluated as \( 2 \ln(2) \), since \( \ln(1) = 0 \).Logarithmic integrations are common, especially with rational functions, and understanding them helps in calculus and real-world applications.

Simplification of Rational Functions

Simplification of rational functions is a vital skill for integrals, especially when dealing with fractions.
In our problem, initially, we have \( \frac{(x+1)^2}{x^2} \). Using polynomial expansion, it reduces to \( 1 + \frac{2}{x} + \frac{1}{x^2} \). This reformulation allows for easy integration.

• The simplification helps in focusing on each separate part, especially when dealing with terms like \( \frac{2}{x} \) or \( \frac{1}{x^2} \).
• You ensure the integral goes faster since each piece uses standard formulas instead of struggling with the original fraction.

By handling rational functions this way, you make complex problems much more approachable and intuitive. Always simplify before integrating.