Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{1}{x^{2}-4 x+9} d x\)

Step-by-Step Solution

Answer

\( \frac{1}{\sqrt{5}} \tan^{-1}\left( \frac{x - 2}{\sqrt{5}} \right) + C \)

Step 1: Identify the Integration Technique

The integrand is a rational function and the denominator is a quadratic expression. The best method to use here is to recognize the form and use a substitution or trigonometric identities if necessary.

Step 2: Complete the Square

To make the integration simpler, let's complete the square for the quadratic expression in the denominator. Start with the quadratic: \[ x^2 - 4x + 9. \]Rewriting it by completing the square:\[ x^2 - 4x + 9 = (x-2)^2 + 5. \]

Step 3: Substitute for Simpler Integration

Substitute \( u = x - 2 \), so that \( du = dx \). This modifies the integral to:\[ \int \frac{1}{u^2 + 5} du. \]

Step 4: Use Trigonometric Substitution

Recognize that the integral \( \int \frac{1}{u^2 + 5} du \) has a form similar to \( \int \frac{1}{a^2 + x^2} dx \), which corresponds to \( \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) + C \). Here, \( a = \sqrt{5} \).

Step 5: Evaluate the Integral

Integrate using the arctangent formula:\[ \int \frac{1}{u^2 + 5} du = \frac{1}{\sqrt{5}} \tan^{-1}\left( \frac{u}{\sqrt{5}} \right) + C. \]

Step 6: Substitute Back to Original Variable

Replace \( u \) with \( x - 2 \) to return to the variable \( x \):\[ \int \frac{1}{x^2 - 4x + 9} dx = \frac{1}{\sqrt{5}} \tan^{-1}\left( \frac{x - 2}{\sqrt{5}} \right) + C. \]

Key Concepts

Rational Function Integration

Integrating rational functions often requires different strategies based on the form of the denominator. A rational function is a ratio of two polynomials where the numerator and the denominator are polynomials. Key steps in integrating rational functions include:

• Identifying if the denominator can be simplified or manipulated using algebraic techniques like completing the square.
• Choosing the right substitution method to transform the integral into a form that is easier to handle.

In the given exercise, the integrand is a rational function with a quadratic denominator. Knowing this allows us to anticipate using techniques such as substitution or a specific formula like the arctangent integration formula.

Trigonometric Substitution

Trigonometric substitution is a useful technique for integrating functions that involve quadratic expressions. It helps when the quadratic resembles the form \( x^2+a^2 \), allowing the use of trigonometric identities to simplify the integration. Here are some general steps:

• Identify the specific form of the quadratic expression, such as \( x^2 + a^2, \) \( a^2 - x^2, \) or \( x^2 - a^2. \)
• Choose the appropriate trigonometric substitution based on the form. For instance, use x = a tan(θ) for \( x^2 + a^2 \), as it aligns with the trigonometric identity \( an^2(θ) + 1 = ext{sec}^2(θ). \)

In the exercise, once the quadratic is in the form \( u^2 + 5, \) a trigonometric substitution is used by recognizing it matches the \( an^{-1} \) integral formula. Hence, understanding this method is crucial for transforming the given integral into a solvable form.

Completing the Square

Completing the square is a technique that reorganizes a quadratic expression into a perfect square plus a constant. This transformation is particularly helpful when dealing with integrals involving quadratic expressions. Here's how you can complete the square:

• Start with an expression of the form \( ax^2 + bx + c. \)
• Divide the linear coefficient, b, by 2, square the result, and add and subtract it within the expression. For example: \( (x^2 - 4x + 9 = (x-2)^2 + 5). \)
• This turns the quadratic into \( (x-d)^2 + e \), making it easier to handle analytically or through further substitution.

In the original problem, completing the square simplifies the expression to \( (x-2)^2 + 5, \) readying it for trigonometric substitution. This step is vital because it converts complex forms into something more manageable for integration.

Arctangent Integration Formula

The arctangent integration formula is a specific result of integration applicable to certain rational functions. With the formula being:\[ \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) + C, \\]it helps in integrating expressions that resemble this format. The steps include:

• Identifying the part of the integrand that can be expressed in the form \( a^2 + x^2. \)
• Recognizing that the integration will result in the arctangent function, adjusting constants as needed.
• Substitute back to the original variable if a substitution was made.

In the given task, after completing the square and substitution, the integral \( \int \frac{1}{u^2 + 5} \, du \) directly uses this formula, leading to the solution. Understanding this formula aids in solving such integrals efficiently.