Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 + x²)] dx

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Identify the form of the integral: \( \int \frac{x}{\sqrt{4 + x^2}} \, dx \). Notice that the expression under the square root is of the form \( a^2 + x^2 \) where \( a = 2 \). This suggests using the trigonometric substitution \( x = 2 \tan(\theta) \).

Compute the differential \( dx \) in terms of \( d\theta \). Since \( x = 2 \tan(\theta) \), then \( dx = 2 \sec^2(\theta) \, d\theta \).

Rewrite the integral in terms of \( \theta \). Substitute \( x = 2 \tan(\theta) \) and \( dx = 2 \sec^2(\theta) \, d\theta \) into the integral, and simplify the square root using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

Simplify the integral expression to a trigonometric integral in terms of \( \theta \). This should reduce to an integral involving \( \tan(\theta) \) and \( \sec(\theta) \) functions that can be integrated using standard trigonometric integral techniques.

After integrating with respect to \( \theta \), substitute back \( \theta = \arctan(\frac{x}{2}) \) to express the answer in terms of \( x \). This completes the evaluation of the integral using trigonometric substitution.

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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 √(a² + x²), substituting x = a tan(θ) transforms the integral into a trigonometric form that is easier to integrate.

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions such as sine, cosine, or tangent. 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 x. This involves using the original substitution and trigonometric identities to rewrite the solution in the original variable's terms.

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.∫ [x / √(4 + x²)] dx