Problem 75 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{x+4}{x^{2}+4} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2} \ln |x^2 + 4| + 2 \tan^{-1}(\frac{x}{2}) + C \).

Step 1: Identify the Integral Type

The given integral is \( \int \frac{x + 4}{x^2 + 4} \, dx \). It appears to be a rational function, and we can consider splitting it into simpler parts if necessary. In this case, we can use a method like partial fraction decomposition or recognize a suitable substitution.

Step 2: Simplifying the Integral

In this case, notice that the numerator can be manipulated to assist in integration. We split the integral as \[ \int \frac{x + 4}{x^2 + 4} \, dx = \int \frac{x}{x^2 + 4} \, dx + \int \frac{4}{x^2 + 4} \, dx. \]

Step 3: Solve the First Integral

For \( \int \frac{x}{x^2 + 4} \, dx \), we use a simple substitution \( u = x^2 + 4 \), hence \( du = 2x \, dx \), or \( x \, dx = \frac{1}{2} du \). So, the integral becomes: \[ \int \frac{x}{x^2 + 4} \, dx = \frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln |u| + C_1 = \frac{1}{2} \ln |x^2 + 4| + C_1. \]

Step 4: Solve the Second Integral

The second part is \( 4 \int \frac{1}{x^2 + 4} \, dx \). This integral closely resembles the standard form \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \). Here, \( a = 2 \), so the integral becomes: \[ 4 \times \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + C_2 = 2 \tan^{-1}\left(\frac{x}{2}\right) + C_2. \]

Step 5: Combine the Results

Combine the results of the integrals from Steps 3 and 4: \[ \int \frac{x + 4}{x^2 + 4} \, dx = \frac{1}{2} \ln |x^2 + 4| + 2 \tan^{-1}\left(\frac{x}{2}\right) + C. \] This is the final solution, combining both constants \( C_1 \) and \( C_2 \) into a single constant \( C \).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a valuable integration technique, particularly useful for breaking down complex rational functions into simpler fractions that are easier to integrate. It works primarily on functions where the degree of the numerator is less than the degree of the denominator. This method is not directly applied to the problem at hand, as the numerator's manipulation led to a simpler integral split.
However, understanding how it works can give insight into solving similar problems. When your rational function does not straightforwardly separate or resemble known integrable forms, consider:

• Factorizing the denominator, if possible.
• Expressing the original function as a sum of simpler fractions based on the factors.
• Integrating each simpler fraction separately.

While it wasn't directly applicable here due to the quadratic nature of the denominator being irreducible over the reals, it's good practice to try decomposition first when dealing with complex rational functions.

Substitution Method

The substitution method is a fantastic tool to simplify integrals by transforming them into an easier form. In our exercise, we used substitution to tackle one part of the integral:

• Recognize a part of the integral as a potential new variable (let's call it "u").
• Differentiate this new variable to find "du", adjusting the differential in the original integral.

For \( \int \frac{x}{x^2 + 4}\, dx \), we chose \( u = x^2 + 4 \) because its derivative \( du = 2x \, dx \) can replace \( x \, dx \) in the integral. This simplification led directly to a natural logarithm, reducing computational hassle. Practice is key when choosing substitutions, as experience will guide your decisions.

Trigonometric Integrals

Trigonometric integrals involve functions like sine, cosine, tangent, and their inverses. In this problem, the term \( 4 \int \frac{1}{x^2 + 4} \, dx \) maps to a standard trigonometric integral form:

• \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \).

This resembles the arctan, or inverse tangent function, showing how trigonometric identities can directly provide solutions to certain integral forms. Here, \( a = 2 \) is a constant factor from the standard form, simplifying the integration into a quick expression. It’s vital to be familiar with these forms, as they make solving problems like this not only faster but less error-prone.

Rational Functions

Rational functions are quotients of polynomials, expressed as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The integral \( \int \frac{x + 4}{x^2 + 4} \, dx \) is a perfect example of a rational function.
To solve such integrals, check whether the degree of the numerator is at least one less than the degree of the denominator. If so, consider partial fraction decomposition or suitable substitutions.

• If the degree of the numerator is equal to or greater than the denominator, perform long division first.
• For this exercise, simplifying and splitting the integral into parts closer to standard forms was the best approach.

Understanding the structure of rational functions helps streamline the problem-solving process, ensuring a systematic approach to integration.