7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
44. ∫ 1/√(16 + 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 the trigonometric substitution u = a * tan(θ). Here, identify a = 4 and u = 2x, so let 2x = 4 * tan(θ).
2. Step 2: Substitute u = 2x = 4 * tan(θ). Compute dx by differentiating: dx = d(2x) = 4 * sec²(θ) dθ.
3. Step 3: Replace √(16 + 4x²) using the trigonometric identity 1 + tan²(θ) = sec²(θ). Substituting, √(16 + 4x²) becomes √(16 * sec²(θ)) = 4 * sec(θ).
4. Step 4: Rewrite the integral in terms of θ using the substitutions: ∫ 1/√(16 + 4x²) dx = ∫ (1 / (4 * sec(θ))) * (4 * sec²(θ) dθ). Simplify the expression to ∫ sec(θ) dθ.
5. Step 5: Integrate ∫ sec(θ) dθ using the standard formula for the integral of sec(θ), which is ln|sec(θ) + tan(θ)| + C. Finally, back-substitute θ using the original substitution 2x = 4 * tan(θ) to express the result in terms of x.

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 tangent, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² + x²), a² - x², or x² - a².

Identifying the Right Triangle

When using trigonometric substitution, it is essential to visualize the relationship between the variable and the trigonometric function through a right triangle. For example, in the integral ∫ 1/√(16 + 4x²) dx, we can set x = 2tan(θ), which leads to a right triangle where the opposite side is 2x and the adjacent side is 4. This helps in determining the appropriate trigonometric identities to apply.

Integration Techniques

After performing the substitution, the integral often requires the application of various integration techniques, such as basic integration rules or further substitutions. In the case of the integral ∫ 1/√(16 + 4x²) dx, after substituting and simplifying, one may need to integrate a trigonometric function, which can involve recognizing standard integral forms or using integration by parts if necessary.