50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0

검증된 단계별 안내

Identify the integral to solve: \(\int x^{n} \cos(ax) \, dx\) where \(a \neq 0\).

Choose functions for integration by parts: let \(u = x^{n}\) (which simplifies when differentiated) and \(dv = \cos(ax) \, dx\) (which integrates easily).

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

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so substitute to get \(\int x^{n} \cos(ax) \, dx = x^{n} \cdot \frac{\sin(ax)}{a} - \int \frac{\sin(ax)}{a} \cdot n x^{n-1} \, dx\).

Factor constants out of the integral to write the reduction formula: \(\int x^{n} \cos(ax) \, dx = \frac{x^{n} \sin(ax)}{a} - \frac{n}{a} \int x^{n-1} \sin(ax) \, dx\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Integration by Parts

Integration by parts is a technique based on the product rule for 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 wisely is crucial to simplify the integral effectively.

Reduction Formulas

Reduction formulas express an integral involving a power or parameter in terms of a similar integral with a lower power or simpler form. They help solve complex integrals step-by-step by reducing the problem to easier cases, often used in integrals involving powers of x and trigonometric functions.

Integration of Trigonometric Functions with Polynomial Factors

Integrals combining polynomials like xⁿ and trigonometric functions such as cos(ax) or sin(ax) require careful application of integration techniques. Recognizing how to handle the polynomial part and the oscillatory trigonometric part is essential, often involving repeated integration by parts to reduce powers.

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0