Question

Evaluate the following integrals, which have irreducible quadratic factors. \(\int \frac{x^{3}+6 x^{2}+3 x+6}{x^{3}+2 x^{2}} d x\)

Short answer

The integral evaluates to \( x + 3\ln|x| - \frac{3}{x} + C \).

Step-by-step solution

Simplify the Rational Function

First, split and simplify the given integrand: \( \frac{x^{3}+6 x^{2}+3 x+6}{x^{3}+2 x^{2}} \). We can separate this into a polynomial part and a proper fraction. The integrand can be expressed as \(1 + \frac{6x^2+3x+6}{x^3+2x^2}\). In this form, the integral is split into two parts: \( \int dx + \int \frac{6x^2+3x+6}{x^3+2x^2} \ dx \).

Partial Fraction Decomposition

The remaining integral, \( \int \frac{6x^2+3x+6}{x^3+2x^2} \, dx \), can be simplified using partial fraction decomposition. We first factorize the denominator: \( x^2(x+2) \). This allows us to express the integrand as \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+2} \).

Solve for Coefficients

To find the constants \(A\), \(B\), and \(C\), equate \(6x^2+3x+6\) to \(A(x^2+2) + Bx(x+2) + Cx^2\). Solve the system of equations generated by matching coefficients of corresponding powers of \(x\). After solving, we get \(A=3\), \(B=3\), and \(C=0\).

Integrate Each Term

Now, integrate each term separately: \[ \int 1 \, dx + \int \frac{3}{x} \, dx + \int \frac{3}{x^2} \, dx. \] The integrals evaluate to \( x + 3\ln|x| - \frac{3}{x} \).

Combine Results

Combine all integrated terms to write the final solution to the original integral. The result is \( x + 3\ln|x| - \frac{3}{x} + C \), where \( C \) is the constant of integration.

Key concepts

Indefinite Integrals

Indefinite integrals are one of the most fundamental concepts in integral calculus. They represent the process of finding a function, known as the antiderivative, that describes the accumulation of quantities whose rate of change is provided by a given function. In simpler terms, when we find an indefinite integral, we are essentially performing the reverse action of differentiation. This helps in constructing a function from its derivative.

• The symbol used for indefinite integrals is \( \int \).
• There is a constant of integration, \( C \), included in the answer, because antiderivatives are not unique and differ by a constant.

Indefinite integrals answer the question: "What is the original function that was differentiated to get this first derivative?" Finding indefinite integrals can appear daunting at first, but by understanding key techniques, like partial fraction decomposition, the task becomes manageable.

Partial Fraction Decomposition

Partial fraction decomposition is a powerful tool used in calculus to simplify complex rational functions into simpler fractions that are easier to integrate. This technique is particularly helpful when dealing with integrals of rational functions — those expressed as the ratio of two polynomials.
Here's how it works:

• The first step is to factor the denominator of the rational function.
• Then, express the original function as the sum of simpler fractions using unknown constants. Each fragment typically corresponds to a factor in the denominator, like \( \frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+2} \).
• Finally, solve for these constants by equating the coefficients of the same powers or by substituting convenient values for the variable.

Partial fraction decomposition allows us to break down a complicated expression into several more digestible integrals, which can be easily solved using standard integration techniques.

Rational Functions

Rational functions are expressions composed of a numerator and denominator that are polynomial functions. Understanding rational functions is essential in calculus because they frequently appear in integration problems. When working with rational functions, there are a few important points to remember:

• The function is expressed as \( \frac{P(x)}{Q(x)} \) where both \(P(x)\) and \(Q(x)\) are polynomials.
• Before integration, ensure the degree (highest power of \(x\)) of the numerator is less than that of the denominator. If not, perform polynomial long division to adjust it into a proper fraction.
• Rational functions can be simplified for integration using techniques like partial fraction decomposition, especially when the denominator factors into linear or simple quadratic terms.

This approach aids in managing and simplifying complex rational expressions, making integration substantially more straightforward.

Polynomial Integration

Polynomial integration is one of the more straightforward forms of integration in calculus, as it generally involves simple algebraic manipulation. Polynomials are sums of terms consisting of a constant multiplied by a variable raised to a non-negative integer power. To integrate polynomials, one typically uses the power rule of integration.
Key steps in polynomial integration include:

• Apply the power rule: For each term in the polynomial \( ax^n \), the integral is \( \frac{a}{n+1}x^{n+1} \) plus a constant of integration, \( C \).
• Sum the integrals of each term to get the total integral of the polynomial.
• Ensure the polynomial is in its simplest form, which might involve initial simplification or use of techniques like polynomial long division.

Understanding polynomial integration simplifies many calculus problems and paves the way to tackling more complex integrals, such as those involving partial fraction decomposition or other advanced methods.