Question

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

Accepted Answer

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.

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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Key Concepts

Partial Fraction Decomposition

Factoring Polynomials

Integration of Rational Functions