Question

Evaluate the following integrals, which have irreducible quadratic factors. \(\int \frac{x}{(x-1)\left(x^{2}+2 x+2\right)^{2}} d x\)

Short answer

\( \int \frac{x}{(x-1)(x^2+2x+2)^2} \, dx = \ln|x^2 + 2x + 2| - \frac{1}{x^2 + 2x + 2} + C \).

Step-by-step solution

Identify the Method

The integral has a rational function with a quadratic polynomial raised to a power in the denominator. This suggests the use of partial fraction decomposition to simplify the integral before attempting to integrate.

Setup the Partial Fraction Decomposition

Set up the form \( \frac{x}{(x-1)(x^2+2x+2)^2} = \frac{A}{x-1} + \frac{Bx + C}{x^2+2x+2} + \frac{Dx + E}{(x^2+2x+2)^2} \). The goal is to find coefficients \(A\), \(B\), \(C\), \(D\), and \(E\) that satisfy this equation for all \(x\).

Expand and Equate the Polynomial

Multiply through by \((x-1)(x^2+2x+2)^2\) to clear the denominators, then equate coefficients of like powers of \(x\) on each side of the resulting equation.

Solve for Coefficients

Solve the resulting system of equations to find the values of \(A\), \(B\), \(C\), \(D\), and \(E\). After calculation, \(A = 0\), \(B = 1\), \(C = 0\), \(D = -1\), and \(E = -2\).

Integrate Each Term

Integrate each term separately using the results from partial fraction decomposition: \[ \int \frac{0}{x-1} \, dx + \int \frac{x}{x^2+2x+2} \, dx - \int \frac{x+2}{(x^2+2x+2)^2} \, dx \].The first integral is zero. To integrate the others, use substitution and standard integral formulas for quadratic denominators.

Solve Each Integral Individually

Integrate each remaining term using substitution: - For \( \int \frac{x}{x^2+2x+2} \, dx \), use the substitution \( u = x^2 + 2x + 2 \), resulting in a natural logarithm function.- For \( \int \frac{x+2}{(x^2+2x+2)^2} \, dx \), use the substitution \( u = x^2 + 2x + 2 \), leading to a result involving \( u^{-2} \).

Combine Results

Combine the results of the integrals into a final expression. After simplification, the integral evaluates to \( \ln|x^2 + 2x + 2| - \frac{1}{x^2 + 2x + 2} + C \), where \( C \) is the integration constant.

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify complex rational expressions, making them easier to integrate. The idea is to express a fraction as a sum of simpler fractions whose individual integrals are easier to evaluate. This is particularly useful when dealing with integrals of rational functions with higher degree polynomials in the denominator.

For example, in the given integral \(\int \frac{x}{(x-1)(x^2+2x+2)^2} \, dx\), we decompose it into a sum of fractions that could be expressed like this:

• \(\frac{A}{x-1}\)
• \(\frac{Bx + C}{x^2+2x+2}\)
• \(\frac{Dx + E}{(x^2+2x+2)^2}\)

The purpose is to find constants A, B, C, D, and E that make the equation true for any \(x\). By multiplying through by the denominator, we eliminate the fractions and can equate coefficients for like powers of \(x\). Solving the system of equations that results gives us the necessary coefficients for integration.

Quadratic Polynomials

Quadratic polynomials are expressions of the form \(ax^2 + bx + c\). They are fundamental in calculus, often appearing as parts of integrals and requiring specific methods to evaluate. In our integral, we deal with \(x^2 + 2x + 2\).

Quadratic polynomials can sometimes be factored into linear terms, but if they are irreducible over the real numbers, as in this exercise, they remain as a single factor raised to a power. Integrals involving such polynomials can often be managed by completing the square or using substitution methods. Recognizing the structure of these polynomials is key in deciding how to proceed with the integration.

• They play role in identifying possible substitutions.
• Help in partial fraction decomposition for reducing complexity.

Integral Evaluation

Integral evaluation is the process of finding the anti-derivative or integral of a function. With the help of substitution and partial fraction decomposition, integrals that look complex at first can be broken down into manageable parts.

In this exercise, once the rational function is decomposed, we evaluate each resulting term separately:

• The integral \(\int \frac{0}{x-1} \, dx\) simplifies to zero.
• The term \(\int \frac{x}{x^2+2x+2} \, dx\) results in a natural log after substitution.
• \(\int \frac{x+2}{(x^2+2x+2)^2} \, dx\) becomes a simple power of \(u\) after substitution, where \(u\) is the quadratic expression.

This step-by-step simplification is key to solving complex integrals neatly and accurately.

Substitution Method

The substitution method is a powerful tool in calculus used to simplify integrals involving complex expressions. This technique involves transforming the variable of integration, reducing the integral into a simpler form that is easier to handle.

In dealing with our integral exercise, the substitution \(u = x^2 + 2x + 2\) is crucial. By differentiating \(u\), we get the derivative \(du = (2x + 2) \, dx\), which helps rewrite the integral in terms of \(u\).

• For the term \(\int \frac{x}{x^2+2x+2} \, dx\), the substitution translates the integral into a form involving \(\ln|u|\).
• For \(\int \frac{x+2}{(x^2+2x+2)^2} \, dx\), the expression becomes \(u^{-2}\), integrating to provide the simplified result.

The key is transforming difficult variables into simpler ones, leading to easier integration and a clear solution expression.