Question

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.
a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Accepted Answer

Step 1: Begin by recalling the reduction formula for integrals involving powers of trigonometric functions. The goal is to express the integral of sin^m(x) in terms of the integral of sin^(m−2)(x). Start with the integral I = ∫ from 0 to π of (sin^m(x)) dx. Step 2: Use integration by parts to simplify the integral. Let u = sin^(m−1)(x) and dv = sin(x) dx. Then, compute du = (m−1)sin^(m−2)(x)cos(x) dx and v = −cos(x). Substitute these into the integration by parts formula: ∫ u dv = uv − ∫ v du. Step 3: Evaluate the boundary terms uv at x = 0 and x = π. Note that sin(0) = sin(π) = 0, so the boundary terms vanish. This simplifies the integral to −∫ from 0 to π of (−cos(x) * (m−1)sin^(m−2)(x)cos(x)) dx. Step 4: Simplify the remaining integral. Combine the $$cos^2(x) $$term and use the Pythagorean identity $$cos^2(x) = 1$$ − $$sin^2(x) to $$express the integral in terms of sin^(m−2)(x). This leads to a recursive relationship between the integrals of sin^m(x) and sin^(m−2)(x). Step 5: Factor out constants and simplify the recursive formula to show that ∫ from 0 to π of (sin^m(x)) dx = (m−1)/m × ∫ from 0 to π of (sin^(m−2)(x)) dx. This completes the proof of the reduction formula.

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.
a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

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125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Risposta video verificata per un problema simile:

Concetti chiave

Reduction Formula

Definite Integral

Wallis Product