Math Problem Statement

∫(8x / (x^3 + x^2 - x - 1)) dx

Solution

The given integral is:

I=∫8xx3+x2−x−1 dxI = \int \frac{8x}{x^3 + x^2 - x - 1} \, dx

Step 1: Factorizing the Denominator

We attempt to factorize the cubic polynomial:

x3+x2−x−1x^3 + x^2 - x - 1

Using the Rational Root Theorem, testing x=1x = 1:

13+12−1−1=1+1−1−1=01^3 + 1^2 - 1 - 1 = 1 + 1 - 1 - 1 = 0

Thus, x−1x - 1 is a factor. Performing polynomial division:

(x3+x2−x−1)÷(x−1)(x^3 + x^2 - x - 1) \div (x - 1)

Using synthetic division:

1 & 1 & 1 & -1 & -1 \\ & & 1 & 2 & 1 \\ \hline & 1 & 2 & 1 & 0 \\ \end{array}$$ So, $$x^3 + x^2 - x - 1 = (x - 1)(x^2 + 2x + 1)$$ Factoring further: $$(x - 1)(x + 1)^2$$ ### Step 2: Partial Fraction Decomposition We write: $$\frac{8x}{(x - 1)(x + 1)^2} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{(x + 1)^2}$$ Multiplying both sides by $$(x - 1)(x + 1)^2$$: $$8x = A(x + 1)^2 + B(x - 1)(x + 1) + C(x - 1)$$ Expanding: $$8x = A(x^2 + 2x + 1) + B(x^2 - 1) + C(x - 1)$$ $$8x = Ax^2 + 2Ax + A + Bx^2 - B + Cx - C$$ $$8x = (A + B)x^2 + (2A + C)x + (A - B - C)$$ Equating coefficients: 1. $$A + B = 0$$ 2. $$2A + C = 8$$ 3. $$A - B - C = 0$$ Solving for $$A, B, C$$: From (1), $$B = -A$$. Substituting into (3): $$A - (-A) - C = 0 \Rightarrow 2A - C = 0$$ Solving (2) and (3): $$2A + C = 8$$ $$2A - C = 0$$ Adding: $$4A = 8 \Rightarrow A = 2$$ Thus, $$B = -2$$. Solving for $$C$$: $$2(2) + C = 8 \Rightarrow 4 + C = 8 \Rightarrow C = 4$$ ### Step 3: Integrating Each Term $$\int \frac{2}{x - 1} \, dx + \int \frac{-2}{x + 1} \, dx + \int \frac{4}{(x + 1)^2} \, dx$$ $$2 \ln |x - 1| - 2 \ln |x + 1| - \frac{4}{x + 1} + C$$ Final answer: $$\ln \left| \frac{(x - 1)^2}{(x + 1)^2} \right| - \frac{4}{x + 1} + C$$ Would you like a detailed explanation of any step? ### Related Questions: 1. How do you perform polynomial long division? 2. What is the general method for solving integrals with partial fractions? 3. How do you identify possible rational roots of a polynomial? 4. How do you integrate functions of the form $$\frac{1}{(x + a)^n}$$? 5. What are some common mistakes in polynomial factorization? **Tip:** Always check for rational roots before attempting polynomial long division!

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Math Problem Analysis

Mathematical Concepts

Integration
Partial Fractions
Polynomial Factorization
Rational Root Theorem

Formulas

Partial fraction decomposition: A/(x - 1) + B/(x + 1) + C/(x + 1)^2
Polynomial division for factorization
Integration of 1/(x - a) as ln|x - a|
Integration of 1/(x + a)^n as -1/(n-1)(x + a)^(n-1) for n > 1

Theorems

Rational Root Theorem
Factorization of Polynomials

Suitable Grade Level

Grades 11-12, College Calculus

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