Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [t / √(4t² − 1)] dt

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Identify the form of the integral and recognize that the expression under the square root, \(4t^{2} - 1\), resembles \(a^{2}t^{2} - b^{2}\), which suggests using a trigonometric substitution for expressions of the form \(\sqrt{a^{2}t^{2} - b^{2}}\).

Set up the substitution by letting \(t = \frac{1}{2} \sec(\theta)\), because \(4t^{2} - 1 = 4\left(\frac{1}{2} \sec(\theta)\right)^{2} - 1 = \sec^{2}(\theta) - 1 = \tan^{2}(\theta)\), which simplifies the square root.

Compute the differential \(dt\) in terms of \(d\theta\): since \(t = \frac{1}{2} \sec(\theta)\), then \(dt = \frac{1}{2} \sec(\theta) \tan(\theta) d\theta\).

Rewrite the integral in terms of \(\theta\) by substituting \(t\), \(dt\), and \(\sqrt{4t^{2} - 1}\) with their trigonometric equivalents, simplifying the integrand accordingly.

Integrate the resulting expression with respect to \(\theta\), then use the inverse trigonometric substitution to express the answer back in terms of \(t\).

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주요 개념

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions by substituting a trigonometric function for the variable. For expressions like √(a²t² − b²), substituting t with a trigonometric function (e.g., t = (b/a) sec θ) transforms the integral into a trigonometric integral that is easier to evaluate.

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions such as secant, tangent, or sine. Understanding how to integrate these functions, including using identities and standard integral formulas, is essential to solve the integral and then revert back to the original variable.

Back-Substitution

Once the integral is evaluated in terms of the trigonometric variable, back-substitution is necessary to express the answer in terms of the original variable. This involves using the original substitution and trigonometric identities to rewrite the solution, ensuring the final answer matches the initial integral's variable.

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.∫ [t / √(4t² − 1)] dt