Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (4 + x²)

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Start by examining the integral \( \int \frac{x^{2}}{4 + x^{2}} \, dx \). Notice that the denominator is \(4 + x^{2}\), which suggests a trigonometric substitution might be useful, but first consider simplifying the integrand by algebraic manipulation.

Rewrite the integrand by expressing \(x^{2}\) in terms of \(4 + x^{2}\): \(x^{2} = (4 + x^{2}) - 4\). Substitute this into the integral to get \(\int \frac{(4 + x^{2}) - 4}{4 + x^{2}} \, dx = \int \left(1 - \frac{4}{4 + x^{2}}\right) \, dx\).

Split the integral into two simpler integrals: \(\int 1 \, dx - 4 \int \frac{1}{4 + x^{2}} \, dx\). This separates the problem into a basic integral and a standard form integral involving \(\frac{1}{a^{2} + x^{2}}\).

Recall the standard integral formula: \(\int \frac{1}{a^{2} + x^{2}} \, dx = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C\). Here, \(a^{2} = 4\), so \(a = 2\). Use this to write the second integral explicitly.

Combine the results: the integral becomes \(\int 1 \, dx - 4 \times \frac{1}{2} \arctan \left( \frac{x}{2} \right) + C\). Simplify the constants and write the final expression in terms of \(x\) and \(\arctan\).

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like a² + x², a² - x², or x² - a² by substituting x with a trigonometric function. This transforms the integral into a trigonometric integral, which is often easier to evaluate. For example, substituting x = 2 tan(θ) can simplify integrals with 4 + x².

Polynomial Division in Integration

When the degree of the numerator is equal to or greater than the denominator, polynomial division can simplify the integrand. Dividing the numerator by the denominator breaks the integral into simpler parts, often a polynomial plus a proper rational function, making the integral easier to solve.

Basic Integration Techniques

Understanding fundamental integration methods such as substitution, recognizing standard integral forms, and using algebraic manipulation is essential. These techniques help in breaking down complex integrals into manageable parts and applying known formulas or substitutions effectively.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ (x² dx) / (4 + x²)