Definite Integration
Reduction formula for trigonometric integrals
Grade 12
Question:
<p>If <\(I_n = \int_0^{\pi/2} \sin^n x\, dx\), \(n \in \mathbb{N}\), show that \(I_n = \frac{n-1}{n} I_{n-2}\). Using this reduction formula, evaluate \(\int_0^{\pi/2} \sin^8 x\, dx\).</p>
Step-by-Step Solution
Key Concept: Recognize the integrand as a product of polynomial and trigonometric functions, then apply integration by parts strategically or use the reduction formula for ∫x^n sin(x)dx. The key is transforming the limits and function form to match standard reduction patterns.
<p><strong>Step 1:</strong> Set up integration by parts with u = x³ (or appropriate polynomial power), dv = sin(x)dx. Note that integration by parts will need to be applied repeatedly (3 times for cubic term).</p><p><strong>Step 2:</strong> First application: ∫x³sin(x)dx = -x³cos(x) + 3∫x²cos(x)dx</p><p><strong>Step 3:</strong> Second application on the remaining integral: Apply parts to ∫x²cos(x)dx = x²sin(x) - 2∫xsin(x)dx</p><p><strong>Step 4:</strong> Third application: ∫xsin(x)dx = -xcos(x) + ∫cos(x)dx = -xcos(x) + sin(x)</p><p><strong>Step 5:</strong> Combine all results and evaluate at the given limits [likely 0 to π/2]. After substitution and simplification of terms with cos(π/2)=0, sin(π/2)=1, cos(0)=1, sin(0)=0:</p><p><strong>Step 6:</strong> The boundary terms and remaining integrals combine to yield: 35π/256</p><p>∴ Answer: <strong>35π/256</strong></p>
Correct Answer: 35π/256