1. What is the Reduction Formula?

The reduction formula for ∫sinⁿx dx is a recursive formula that expresses the integral of sinⁿx in terms of the integral of sinⁿ⁻²x. This allows for step-by-step reduction of the exponent until reaching a base case that can be easily integrated.

2. How Does the Formula Work?

The reduction formula is:

Where:

• \( n \) — Positive integer exponent (n ≥ 2)
• \( x \) — Variable in radians
• \( \sin^{n-1} x \) — Sine function raised to power (n-1)
• \( \cos x \) — Cosine function

Explanation: The formula reduces the power of the sine function by 2 in each iteration, making complex integrals manageable through repeated application.

3. Applications of Reduction Formulas

Details: Reduction formulas are essential tools in calculus for solving integrals of trigonometric functions, especially when dealing with higher powers. They simplify complex integration problems by breaking them down into simpler components.

4. Using the Calculator

Tips: Enter a positive integer for n (n ≥ 2) and a value for x in radians. The calculator will show the reduction step for the given parameters.

5. Frequently Asked Questions (FAQ)

Q1: What are the base cases for this reduction formula?
A: The base cases are ∫sin⁰x dx = ∫1 dx = x + C and ∫sin¹x dx = -cosx + C.

Q2: Can this formula be used for even and odd powers?
A: Yes, the reduction formula works for both even and odd positive integer powers of sine.

Q3: How many reduction steps are needed?
A: For sinⁿx, you need ⌊n/2⌋ reduction steps to reach the base case.

Q4: Are there similar reduction formulas for cosine?
A: Yes, there's a similar reduction formula for ∫cosⁿx dx with a different structure.

Q5: When should I use reduction formulas vs other integration methods?
A: Use reduction formulas when dealing with powers of trigonometric functions, especially when substitution or other methods become cumbersome.