Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)

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Identify the integral to solve: \(\int \frac{x}{25 + 4x^{2}} \, dx\).

Recognize that the denominator is a quadratic expression of the form \(a^{2} + (bx)^{2}\), which suggests a trigonometric substitution or a simpler method might be effective.

Consider using a substitution method: let \(u = 25 + 4x^{2}\). Then compute \(du\) in terms of \(dx\) and \(x\).

Calculate \(du\): since \(u = 25 + 4x^{2}\), then \(\frac{du}{dx} = 8x\), so \(du = 8x \, dx\). This allows you to express \(x \, dx\) in terms of \(du\).

Rewrite the integral in terms of \(u\) using the substitution, then integrate with respect to \(u\). After integrating, substitute back to express the answer in terms of \(x\).

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Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate 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 solve. For example, substituting x = (a/2)tan(θ) can simplify integrals with terms like 25 + 4x².

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting u be a function of x, the integral can be rewritten in terms of u, making it easier to evaluate. This method is useful when the integral contains a function and its derivative, as it can reduce the integral to a basic form.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. This method applies when the denominator can be factored into linear or quadratic terms. Although not always necessary, it is a useful alternative when trigonometric substitution is complicated or not straightforward.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ (x dx) / (25 + 4x²)