In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ 1 / (x⁴ + x) dx

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Start by factoring the denominator of the integrand. The denominator is \(x^{4} + x\), which can be factored by taking out the common factor \(x\): \[x^{4} + x = x(x^{3} + 1)\]

Next, factor the cubic term \(x^{3} + 1\) using the sum of cubes formula: \[a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\] Here, \(a = x\) and \(b = 1\), so \[x^{3} + 1 = (x + 1)(x^{2} - x + 1)\]

Rewrite the integrand using the full factorization: \[\frac{1}{x^{4} + x} = \frac{1}{x(x + 1)(x^{2} - x + 1)}\]

Set up the partial fraction decomposition for the integrand: \[\frac{1}{x(x + 1)(x^{2} - x + 1)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{Cx + D}{x^{2} - x + 1}\] where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Multiply both sides by the denominator \(x(x + 1)(x^{2} - x + 1)\) to clear the fractions, then equate coefficients of corresponding powers of \(x\) to form a system of equations. Solve this system to find the values of \(A\), \(B\), \(C\), and \(D\). Once found, integrate each term separately to evaluate the integral.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a complex rational function as a sum of simpler fractions. This method is especially useful for integrating rational functions by breaking them into terms that are easier to integrate individually.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For partial fractions, factoring the denominator into linear and irreducible quadratic factors is essential to set up the correct form of the decomposition.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts via partial fractions. Each simpler fraction corresponds to a standard integral form, such as logarithmic or arctangent functions, facilitating the evaluation of the integral.

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.∫ 1 / (x⁴ + x) dx