Problem 14 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{x^{2}+4 x}{x^{3}+6 x^{2}+5} d x $$

Step-by-Step Solution

Answer

The integral \(\int{ \frac{x^{2}+4x}{x^{3}+6x^{2}+5} } dx\) will be found by first splitting the fraction into simpler fractions and then finding the integral of each resulting fraction.

Step 1: Polynomial Long Division

First, perform polynomial long division on the fraction \(\frac{x^{2}+4x}{x^{3}+6x^{2}+5}\). This is done to separate the fraction into simpler fractions.

Step 2: Identify Simple Fractions

After performing long division, the result \(F(x)\) can be written as \(F(x)= \frac{x^{2}+4x}{x^{3}+6x^{2}+5} = \frac{A}{x+1} + \frac{Bx+C}{x^{2}+5}\), where A, B and C are constants. Identify the values of A, B and C by multiplying both sides by \((x+1)(x^{2}+5)\) and comparing coefficients of powers of x.

Step 3: Integral Calculation

Calculate the integrals \(\int{ \frac{A}{x+1} } dx\) and \(\int{ \frac{Bx+C}{x^{2}+5} } dx\) separately using standard integral results.

Key Concepts

Polynomial Long Division

Polynomial long division is a technique used to divide one polynomial by another, yielding a quotient and a remainder, much like long division with numbers. This method is especially useful when dealing with rational expressions where the degree of the numerator is equal to or greater than the degree of the denominator.

To perform polynomial long division:

• Write the dividend (numerator) and the divisor (denominator) in descending power order.
• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Repeat this process with the new, lower degree polynomial that results, until the degree of the remainder is less than the degree of the divisor.

In the context of integration, long division simplifies the given rational expression into a form that can be more easily integrated using standard techniques, aiming for a result that combines a polynomial and a proper fractional part if any remainder is present.

Partial Fraction Decomposition

Partial fraction decomposition is an algebraic technique that breaks down a complex rational expression into simpler fractions that are easier to integrate. It is particularly useful for integrating rational functions where the denominator can be factored into linear or irreducible quadratic polynomials. Here's a simplified approach to partial fraction decomposition:

• Express the rational function as a sum of fractions with unknown coefficients.
• The denominators are the factors of the original denominator.
• Clear the fractions by multiplying through by the common denominator to get an equation in terms of the original polynomial.
• Solve for the unknown coefficients by equating the coefficients of corresponding powers of x on both sides.

Once we have the values of the coefficients, we substitute them back into the decomposed fractions. The integral of the original function can now be broken down into the sum of simpler integrals.

Integration Techniques

Integration can often be challenging, but various techniques make it manageable. The most fundamental method is the direct integration of basic functions using standard formulas. When direct integration is not possible, such as with composite or more complex functions, other techniques like substitution or integration by parts are employed.

For rational functions, depending on the structure of the function after long division and partial fraction decomposition, one might integrate:

• Polynomials term-by-term directly.
• Inverses of linear functions using the natural logarithm, \( \ln |x| \).
• Rational functions with quadratic denominators using trigonometric substitutions or completing the square.

The key to successful integration is selecting the method that simplifies the integral most effectively, whether it's a standard antiderivative, substitution, parts, or a combination of these techniques.