Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ x² sin(x) dx

검증된 단계별 안내

Identify the parts of the integral for integration by parts. Let \( u = x^2 \) and \( dv = \sin(x) \, dx \).

Compute the derivatives and antiderivatives needed: \( du = 2x \, dx \) and \( v = -\cos(x) \) because the integral of \( \sin(x) \) is \( -\cos(x) \).

Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions to get \( \int x^2 \sin(x) \, dx = -x^2 \cos(x) - \int -\cos(x) \cdot 2x \, dx \).

Simplify the integral: \( -x^2 \cos(x) + 2 \int x \cos(x) \, dx \). Now, you need to evaluate \( \int x \cos(x) \, dx \) using integration by parts again.

For the new integral \( \int x \cos(x) \, dx \), set \( u = x \) and \( dv = \cos(x) \, dx \), then repeat the integration by parts steps to solve it.

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 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 wisely simplifies the problem, especially when integrating products like x² sin(x).

Choosing u and dv

Selecting which part of the integrand to assign as u and which as dv is crucial. Typically, u is chosen as a function that simplifies when differentiated (like polynomials), and dv is chosen as a function that is easy to integrate (like trigonometric functions). This choice reduces the complexity of the resulting integral.

Repeated Application of Integration by Parts

When the integral involves powers of x multiplied by trigonometric functions, integration by parts may need to be applied multiple times. Each application reduces the power of x until the integral becomes straightforward. This iterative process is essential for solving integrals like ∫ x² sin(x) dx.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ x² sin(x) dx