In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx

Verified step by step guidance

1. Identify the parts for integration by parts: let \(u = x^n\) and \(dv = \sin(x) \, dx\).
2. Compute the derivatives and integrals needed: \(du = n x^{n-1} \, dx\) and \(v = -\cos(x)\) (since \(\frac{d}{dx}(-\cos(x)) = \sin(x)\)).
3. Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(\int x^n \sin(x) \, dx = -x^n \cos(x) - \int (-\cos(x)) (n x^{n-1}) \, dx\).
4. Simplify the integral expression: \(-x^n \cos(x) + n \int x^{n-1} \cos(x) \, dx\).
5. This establishes the reduction formula: \(\int x^n \sin(x) \, dx = -x^n \cos(x) + n \int x^{n-1} \cos(x) \, dx\).

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv appropriately is crucial to simplify the integral effectively.

Reduction Formula

A reduction formula expresses an integral involving a parameter (like n) in terms of a similar integral with a lower parameter value. It helps solve integrals recursively by breaking them down step-by-step, making complex integrals manageable through repeated application.

Handling Powers of x in Integration

When integrating expressions like x^n multiplied by trigonometric functions, powers of x are reduced systematically. Differentiating x^n reduces the power by one, which is essential in forming the reduction formula and simplifying the integral through repeated integration by parts.