7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²) dx

Verified step by step guidance

1. Step 1: Factorize the denominator x⁴ + x². Notice that x⁴ + x² can be factored as x²(x² + 1). This simplifies the integral to ∫ 1/(x²(x² + 1)) dx.
2. Step 2: Use partial fraction decomposition to break the fraction into simpler terms. Write 1/(x²(x² + 1)) as A/x² + B/(x² + 1), where A and B are constants to be determined.
3. Step 3: Solve for A and B by equating the numerators. Multiply through by the denominator x²(x² + 1) and solve the resulting equation for A and B.
4. Step 4: Rewrite the integral using the partial fractions obtained. The integral becomes ∫ A/x² dx + ∫ B/(x² + 1) dx.
5. Step 5: Evaluate each term separately. For ∫ A/x² dx, use the power rule for integration, and for ∫ B/(x² + 1) dx, recognize it as a standard integral involving arctangent. Combine the results to express the solution.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be understood as the reverse process of differentiation. In this context, evaluating the integral ∫ 1/(x⁴ + x²) dx requires techniques such as substitution or partial fraction decomposition to simplify the integrand.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function into simpler fractions that are easier to integrate. This technique is particularly useful when the integrand is a rational function, like 1/(x⁴ + x²). By expressing the function as a sum of simpler fractions, we can integrate each term individually, making the overall integration process more manageable.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. In the case of the integrand 1/(x⁴ + x²), recognizing that it can be factored as 1/(x²(x² + 1)) is crucial. This factorization simplifies the integration process and allows for the application of techniques such as partial fraction decomposition, facilitating the evaluation of the integral.