Problem 43 - Calculus Single Variable

Question

Calculate the given integral. \(\int \frac{8 x}{x^{5}-x^{4}-x+1} d x\)

Step-by-Step Solution

Answer

The integral can be solved by first factoring the denominator and utilizing partial fraction decomposition to break it into easier terms to integrate.

Step 1: Analyzing the Integral

Look at the integral: \( \int \frac{8x}{x^5 - x^4 - x + 1} \, dx \). Notice that the degree of the polynomial in the denominator is higher than the degree of the polynomial in the numerator.

Step 2: Partial Fraction Decomposition

Assume we can express the integrand as a sum of partial fractions, but this method requires factoring the denominator, \(x^5 - x^4 - x + 1\). Start by checking for factorization possibilities or root detection methods, like the Rational Root Theorem, if applicable.

Step 3: Factoring the Denominator

Use algebra or a factoring tool to attempt to factor the polynomial \(x^5 - x^4 - x + 1\). Suppose it factors as \((x + a)(x + b)(x^3 + cx^2 + dx + e)\). Verify the factorization is correct with polynomial long division or substitution.

Step 4: Set Up Partial Fractions

If the denominator factors successfully, set up a partial fraction decomposition of the integrand: \(\frac{8x}{(x + a)(x + b)(x^3 + cx^2 + dx + e)} = \frac{A}{x + a} + \frac{B}{x + b} + \frac{Cx^2 + Dx + E}{x^3 + cx^2 + dx + e}\).

Step 5: Solve for Coefficients

Multiply through by the common denominator and equate the coefficients from both sides of the equation. Solve the resulting system of equations to find the constants \(A, B, C, D, E\).

Step 6: Integrate Each Term

Integrate each part of the partial fraction decomposition separately. Use \(\int \frac{1}{x +a} \, dx = \ln|x + a| + C\), and for quadratic terms, consider logarithmic or arctangent integrals depending on the specific form.

Step 7: Combine Results

Combine the integrated results to form the solution to the original integral. Don't forget to include the constant of integration \(C\) in the final result.

Key Concepts

Polynomial Long Division

Polynomial long division is a method similar to numerical long division. It helps divide one polynomial by another when simplification is needed, as in the case of partial fraction decomposition.
Start by arranging the terms of both the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by) in descending order of their degrees. For each step:

• Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this term, and subtract the result from the dividend.
• Bring down the next term from the dividend and repeat the process until you reach a remainder that is of a lower degree than the divisor.

This division process allows you to express the original fraction as a combination of a quotient and a remainder. It is especially useful in simplifying complex fractions in integral calculus or when the polynomial needs to be rewritten in a simpler form.

Rational Root Theorem

The Rational Root Theorem is a useful tool for finding potential rational roots of a polynomial equation. It states that any possible rational root of the polynomial equation \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 = 0\) must be a factor of the constant term \(a_0\) divided by a factor of the leading coefficient \(a_n\).
This means if you're looking for rational solutions of \(x^5 - x^4 - x + 1 = 0\), you should:

• List all possible factors of the constant term, here 1, which are \(\pm 1\).
• List all possible factors of the leading coefficient, here 1, which again are \(\pm 1\).
• Identify all potential rational roots by dividing factors of the constant term by factors of the leading term, which results here in the possible roots being \(\pm 1\).

Testing these values in the polynomial function helps confirm if they are indeed roots. Detecting rational roots can simplify further factorization of polynomials, leading to effective partial fraction decomposition for integration.

Integration Techniques

In calculus, integration techniques are methods used to find integrals or antiderivatives. Depending on the form of the function, different techniques need to be applied to simplify the integration process.
Some commonly used techniques include:

• Substitution: Useful when the integral contains a composite function. By making a substitution, you can transform the integral into a simpler form.
• Integration by Parts: An application of the product rule for differentiation. Ideal for products of functions, like polynomials against logarithmic or exponential functions.
• Partial Fraction Decomposition: Great for rational functions where the degree of the numerator is less than the degree of the denominator. After factoring the denominator, break the fraction into simpler fractions, then integrate each one separately.
• Trigonometric Integrals: Use when dealing with integrands involving trigonometric functions. This might involve converting to different trigonometric identities for simplification.

The choice of technique is vital and largely depends on the form of the function you're trying to integrate, ensuring the use of the simplest and most effective method.

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of its factors. This is crucial for simplification, especially in the context of solving equations or performing integration using partial fraction decomposition.
For a polynomial like \(x^5 - x^4 - x + 1\), factorization can be approached using several strategies:

• Rational Root Theorem: Begin by identifying any potential rational roots as a starting point for exploring further factorization.
• Synthetic Division: Test potential roots by performing synthetic division. If no remainder exists, the divisor is a factor.
• Grouping: Sometimes, polynomials can be factored by grouping terms to find a common factor among subsets of terms.
• Use of Algebraic Identities: Certain identities may simplify the factorization process, such as the difference of squares or sum and difference of cubes.

Once the polynomial is successfully factored, the decomposition follows naturally, making subsequent integration through partial fractions more feasible.