91Ó°ÊÓ

Answer: A reduction formula is a mathematical expression that helps to evaluate an integral by simplifying it into a combination of simpler functions and another integral involving a lower degree parameter. Essentially, it helps to evaluate a sequence of successive integrals by breaking them down into a simpler form. To derive a reduction formula step by step, follow these steps: 1. Choose a generic integral expression with a positive integer parameter, for example: I_n = ∫ sin^n(x) dx, where n is a positive integer. 2. Apply the integration by parts formula ( ∫ u dv = uv - ∫ v du) by selecting appropriate u and dv from the given expression. In our example, u = sin^n(x) and dv = dx, resulting in du = n·sin^{n-1}(x)cos(x) dx and v = x. 3. Substitute u, v, du, and dv into the integration by parts formula: I_n = sin^n(x) · x - ∫ x · n·sin^{n-1}(x)cos(x) dx. 4. Simplify the expression and apply trigonometric identity sin^2(x) + cos^2(x) = 1: I_n = sin^n(x) · x - n ∫ x · sin^{n-1}(x)(1 - sin^2(x)) dx. 5. Simplify the last expression by expanding and separating integrals: I_n = sin^n(x) · x - n ∫ x · sin^{n-1}(x) dx + n ∫ x · sin^{n+1}(x) dx. Now, the reduction formula for the integral of the sine function with a positive integer exponent is derived, and it can be used to break down and evaluate the integral for any positive integer n.