7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves a square root of the form √(a² - u²), which suggests using a trigonometric substitution. Specifically, use the substitution u = (a/b)sin(θ), where a² = 9 and b² = 4. Here, let x = (3/2)sin(θ), so dx = (3/2)cos(θ)dθ.
2. Step 2: Substitute x = (3/2)sin(θ) into the square root expression √(9 - 4x²). This becomes √(9 - 4((3/2)sin(θ))²). Simplify the expression inside the square root using trigonometric identities.
3. Step 3: Simplify the square root. After substitution and simplification, the square root becomes √(9 - 9sin²(θ)), which simplifies further to √(9(1 - sin²(θ))). Using the Pythagorean identity 1 - sin²(θ) = cos²(θ), the square root becomes 3cos(θ).
4. Step 4: Rewrite the integral in terms of θ. The integral ∫√(9 - 4x²)dx becomes ∫3cos(θ) * (3/2)cos(θ)dθ, which simplifies to ∫(9/2)cos²(θ)dθ.
5. Step 5: Use the trigonometric identity cos²(θ) = (1 + cos(2θ))/2 to simplify the integral further. Substitute this identity into the integral and proceed to integrate term by term. After integration, convert back to the original variable x using the substitution x = (3/2)sin(θ).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as sine or cosine, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² - x²), which can be simplified using the identity sin²(θ) + cos²(θ) = 1.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that states sin²(θ) + cos²(θ) = 1. This identity is crucial when performing trigonometric substitutions, as it allows for the conversion between different trigonometric functions and helps simplify expressions involving square roots. For example, if x is substituted with a, sin(θ) can be expressed in terms of x, facilitating the integration process.

Integral Evaluation

Integral evaluation is the process of finding the antiderivative of a function, which represents the area under the curve of that function. In the context of trigonometric substitution, once the substitution is made and the integral is simplified, standard integration techniques can be applied. After evaluating the integral, it is essential to revert back to the original variable to express the final answer in terms of x.