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 √(x² - 4) dx

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Identify the integral to solve: \(\int x \sqrt{x^{2} - 4} \, dx\).

Recognize that the integrand contains a square root of the form \(\sqrt{x^{2} - a^{2}}\), which suggests using a trigonometric substitution where \(x = 2 \sec \theta\) because \(a = 2\).

Substitute \(x = 2 \sec \theta\), then compute \(dx = 2 \sec \theta \tan \theta \, d\theta\). Also, rewrite the square root: \(\sqrt{x^{2} - 4} = \sqrt{4 \sec^{2} \theta - 4} = 2 \tan \theta\).

Rewrite the integral entirely in terms of \(\theta\): replace \(x\), \(\sqrt{x^{2} - 4}\), and \(dx\) with their trigonometric expressions, then simplify the integrand before integrating with respect to \(\theta\).

After integrating with respect to \(\theta\), use the original substitution \(x = 2 \sec \theta\) to rewrite the answer back in terms of \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(x² - a²), √(a² - x²), or √(x² + a²). By substituting x with a trigonometric function (e.g., x = a sec θ for √(x² - a²)), the integral transforms into a trigonometric integral that is easier to evaluate.

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. After choosing an appropriate substitution (often related to the expression inside the root), the integral is rewritten in terms of a new variable, making it easier to integrate and then back-substitute to the original variable.

Evaluating Integrals Involving Radicals

Integrals containing radicals like √(x² - a²) often require special techniques such as trigonometric substitution or algebraic manipulation. Recognizing the form of the radical helps determine the best approach to simplify and evaluate the integral effectively.

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Textbook Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dθ / √(2θ - θ²))

Textbook Question

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx

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Textbook Question

Evaluate the integrals in Exercises 39–54.

∫ 1 / ((x¹/³ - 1)√x) dx

(Hint: Let x = u⁶.)

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Textbook Question

Evaluate the integrals in Exercises 53–58.