90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (x² / (x⁴ + x²)) dx

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Step 1: Simplify the integrand. Notice that the denominator can be factored as x²(x² + 1). Rewrite the integrand as (x² / (x²(x² + 1))) = 1 / (x² + 1).

Step 2: Recognize that the simplified integrand, 1 / (x² + 1), resembles the derivative of the arctangent function. Recall that the derivative of arctan(x) is 1 / (1 + x²).

Step 3: Adjust the integrand to match the standard form. In this case, the integrand is already in the form 1 / (x² + 1), which corresponds directly to the derivative of arctan(x).

Step 4: Integrate the simplified function. The integral of 1 / (x² + 1) is arctan(x) + C, where C is the constant of integration.

Step 5: Write the final expression for the indefinite integral as ∫ (x² / (x⁴ + x²)) dx = arctan(x) + C.

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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation, and it is fundamental in calculus for solving problems related to area under curves and accumulation functions.

Integration Techniques

Various techniques are employed to solve integrals, especially when dealing with complex functions. Common methods include substitution, integration by parts, and partial fraction decomposition. For the given integral, recognizing the structure of the integrand can guide the choice of technique, such as simplifying the expression or breaking it into simpler fractions for easier integration.

Rational Functions

A rational function is a ratio of two polynomials. In the context of integration, understanding the behavior of rational functions is crucial, particularly in simplifying the integrand. The integral provided involves a rational function, and techniques like partial fraction decomposition can be used to express it in a form that is easier to integrate, allowing for a clearer path to the solution.

90–103. Indefinite integrals Determine the following indefinite integrals.∫ (x² / (x⁴ + x²)) dx