Math Problem Statement

Evaluate the integral: ∫(π/6 to π/3) [(x - sin(x)) / (x^2 * sin^n(x))] * cot^(n-2)(x) dx

Solution

To solve the given integral:

π/6π/3xsinxx2sinnxcotn2xdx\int_{\pi/6}^{\pi/3} \frac{x - \sin x}{x^2 \sin^n x} \cdot \cot^{n-2} x \, dx

Here is the step-by-step approach:

  1. Simplify the Expression: Analyze the components of the integrand. Notice the use of sinx\sin x, cotx=cosxsinx\cot x = \frac{\cos x}{\sin x}, and the powers nn and n2n-2. The goal is to rewrite the integrand in a form that is easier to integrate.

  2. Expand and Use Substitution: If (n

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Math Problem Analysis

Mathematical Concepts

Definite Integration
Trigonometric Functions
Substitution Method

Formulas

cot(x) = cos(x) / sin(x)
Integration by substitution
Properties of definite integrals

Theorems

Trigonometric Identities
Integration techniques for rational and trigonometric functions

Suitable Grade Level

Undergraduate Mathematics or Advanced Placement Calculus