76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

Verified step by step guidance

1. Step 1: Begin by substituting u = √x. This implies that x = u² and dx = 2u du. Rewrite the integral in terms of u.
2. Step 2: Substitute x = u² into the expression under the square root: √(x - x²) becomes √(u² - u⁴).
3. Step 3: Factorize the expression under the square root: u² - u⁴ = u²(1 - u²). Thus, √(u² - u⁴) = u√(1 - u²).
4. Step 4: Replace dx with 2u du and simplify the integral: ∫(1/√(x - x²)) dx becomes ∫(1/(u√(1 - u²)) * 2u du). Cancel out u in the numerator and denominator.
5. Step 5: The integral simplifies to ∫(2/√(1 - u²)) du. Recognize this as the standard integral for the arcsine function, ∫(1/√(1 - u²)) du = sin⁻¹(u) + C. Substitute back u = √x to get the final result: 2 sin⁻¹√x + C.

Key Concepts

Substitution in Integration

Substitution is a technique used in integration to simplify the integrand. By replacing a variable with another expression, the integral can often be transformed into a more manageable form. In this case, substituting u = √x allows us to express the integral in terms of u, making it easier to evaluate.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find angles when given the value of a trigonometric function. In this problem, the result involves sin⁻¹(√x), which indicates that the integral's solution relates to the angle whose sine is √x. Understanding these functions is crucial for interpreting the final result of the integral.

Definite vs. Indefinite Integrals

Indefinite integrals represent a family of functions and include a constant of integration (C) in their results. The integral in this question is indefinite, meaning it does not have specified limits of integration. Recognizing the difference between definite and indefinite integrals is essential for correctly interpreting the outcome of integration problems.