Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ p⁴ e^(-p) dp

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Identify the integral to solve: \(\int p^{4} e^{-p} \, dp\).

Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).

Choose \(u\) and \(dv\) wisely. Let \(u = p^{4}\) (a polynomial) and \(dv = e^{-p} \, dp\) (an exponential function).

Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 4p^{3} \, dp\), and integrate \(dv\) to get \(v = -e^{-p}\).

Apply the integration by parts formula: substitute \(u\), \(v\), \(du\), and \(dv\) into \(\int u \, dv = uv - \int v \, du\), then simplify and repeat integration by parts on the resulting integral as needed until the polynomial term is fully reduced.

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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 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 simplifies the problem, especially when one function becomes simpler upon differentiation.

Polynomial and Exponential Functions

Understanding the behavior of polynomial functions like p⁴ and exponential functions like e^(-p) is crucial. Polynomials reduce in degree when differentiated, while exponentials maintain their form. This interplay makes integration by parts effective for integrals involving their products.

Repeated Application of Integration by Parts

When integrating products involving high-degree polynomials, integration by parts often needs to be applied multiple times. Each iteration reduces the polynomial's degree by one, gradually simplifying the integral until it becomes straightforward to evaluate.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ p⁴ e^(-p) dp