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

Verified step by step guidance

1. Identify the form of the integral: the integrand contains \( \sqrt{4 - x^2} \), which suggests using the trigonometric substitution \( x = 2 \sin(\theta) \) because \( 4 - x^2 = 4 - 4\sin^2(\theta) = 4\cos^2(\theta) \).
2. Compute the differential \( dx \) in terms of \( d\theta \): since \( x = 2 \sin(\theta) \), then \( dx = 2 \cos(\theta) d\theta \).
3. Rewrite the integral in terms of \( \theta \): substitute \( x = 2 \sin(\theta) \), \( dx = 2 \cos(\theta) d\theta \), and \( \sqrt{4 - x^2} = 2 \cos(\theta) \) into the integral \( \int \frac{x}{\sqrt{4 - x^2}} dx \).
4. Simplify the integral expression after substitution: the integral becomes \( \int \frac{2 \sin(\theta)}{2 \cos(\theta)} \times 2 \cos(\theta) d\theta \), which simplifies to \( \int 2 \sin(\theta) d\theta \).
5. Integrate with respect to \( \theta \): find \( \int 2 \sin(\theta) d\theta \), then substitute back \( \theta = \arcsin(\frac{x}{2}) \) to express the answer in terms of \( x \).

Key Concepts

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 sin(θ) transforms the integral into a trigonometric form that is easier to evaluate.

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions such as sine and cosine. Understanding how to integrate these functions, including using identities and basic 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 used to rewrite the answer in terms of the original variable x. This involves using the inverse trigonometric functions or right triangle relationships derived from the substitution.