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

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Identify the parts for integration by parts: let \(u = x^n\) and \(dv = \sin(x) \, dx\).

Compute the derivatives and integrals needed: \(du = n x^{n-1} \, dx\) and \(v = -\cos(x)\) (since \(\frac{d}{dx}(-\cos(x)) = \sin(x)\)).

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(\int x^n \sin(x) \, dx = -x^n \cos(x) - \int (-\cos(x)) (n x^{n-1}) \, dx\).

Simplify the integral expression: \(-x^n \cos(x) + n \int x^{n-1} \cos(x) \, dx\).

This establishes the reduction formula: \(\int x^n \sin(x) \, dx = -x^n \cos(x) + n \int x^{n-1} \cos(x) \, dx\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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