Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{5-x}{x^{2}-x(\pi+4)+4 \pi} d x\)

Step-by-Step Solution

Answer

Complete the square and use partial fraction decomposition to facilitate integration.

Step 1: Identify the Quadratic Denominator

The given integral is \( \int \frac{5-x}{x^{2}-x(\pi+4)+4\pi} \ dx \). The first step is to observe the quadratic part of the denominator: \( x^{2} - x(\pi + 4) + 4\pi \). This quadratic expression may be set up for factoring or completing the square.

Step 2: Complete the Square

To facilitate partial fraction decomposition, rewrite the quadratic in the form \((x - h)^2 + k\). Complete the square for \(x^{2} - x(\pi + 4)\):\[x^2 - x(\pi + 4) = (x - \frac{(\pi + 4)}{2})^2 - (\frac{(\pi + 4)}{2})^2\]This results in the expression:\[(x - \frac{(\pi + 4)}{2})^2 + (4\pi - (\frac{(\pi + 4)}{2})^2)\]Solve for \(k\) to get the completed square.

Step 3: Factor Denominator or Confirm Irreducibility

For partial fraction decomposition, the quadratic form should be a product of linear factors, or it's irreducible. Confirm if the expression \((x - \frac{(\pi + 4)}{2})^2 + k\) is irreducible if \(k > 0\). Since we are dealing with an arbitrary expression, assume it is irreducible.

Step 4: Set Up Partial Fraction Decomposition

Assuming the denominator is irreducible, we express \(\frac{5-x}{(x - \frac{(\pi + 4)}{2})^2 + k}\) as:\[\frac{5-x}{(x - \frac{(\pi + 4)}{2})^2 + k} = \frac{Ax + B}{((x - \frac{(\pi + 4)}{2})^2 + k)}\]This expression implies potential identification of coefficients through strategic selection or components equivalence. The given problem does not easily allow for linear factoring, thus consider completing the square led steps.

Step 5: Perform Integration by Substitution or Completion

Assuming the irreducible form persisted, use trigonometric substitutions or lookup integration techniques like arctangent and logarithmic solutions for completing the square forms. If directly substituted, set \(u = x - \frac{(\pi + 4)}{2}\), and transform the integral for easier computation.

Step 6: Compute the Integral

After final substitution, determine:\[\int \frac{5-x}{(x - h)^2 + k} \, dx\]Solve by evaluating an expression analogous to the arctangent function given quadratic irreducibility. Use appropriate antiderivatives for composite expressions.

Key Concepts

Integral Calculus

Integral calculus deals with the concept of integration, which is used to calculate areas under curves, among other applications. Integration is essentially the reverse process of differentiation.
In this problem, we are tasked with evaluating an integral using

• Partial Fraction Decomposition: This is a technique to break down complex rational functions into simpler parts that are easier to integrate.
• Substitution: This method can simplify the integration process by changing variables.

Here, the integral expression \[\int \frac{5-x}{x^{2} - x(\pi+4) + 4\pi} \, dx\]needs to be tackled through these methods to yield a solution that renders the calculation manageable.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. A general form can be written as \(ax^2 + bx + c\). In this exercise, the expression \[x^2 - x(\pi + 4) + 4\pi\]is integral to simplifying our integrand.

Key components of quadratic expressions:

• The squared term \(x^2\) signifies a degree two polynomial.
• Linear term including constants, found in expressions involving \(x\).
• Constant term, a standalone number without variables.

Recognizing these components aids in strategic mathematical techniques like completing the square.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This process is particularly useful in the integration of rational expressions with quadratic denominators.

For a quadratic expression like our \[x^2 - x(\pi + 4) + 4\pi\]we aim to express it in the form \((x - h)^2 + k\). This transformation involves strategic manipulation to reveal a perfect square plus a constant:

• Start with \(x^2 - x(\pi + 4)\).
• Identify the necessary constant to add and subtract for perfect square formation.
• The square is achieved by dividing the coefficient of \(x\) in half, squaring it, and using it for completing the square.

By completing the square, we make irreducible quadratic expressions more tractable for integration, often involving methods like substitution or recognizing standard integral forms.