Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (r² + r + 1) e^r dr

검증된 단계별 안내

Identify the integral to solve: \(\int (r^{2} + r + 1) e^{r} \, dr\).

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

Choose parts for \(u\) and \(dv\): let \(u = r^{2} + r + 1\) (a polynomial) and \(dv = e^{r} \, dr\) (an exponential function).

Compute \(du\) and \(v\): differentiate \(u\) to get \(du = (2r + 1) \, dr\), and integrate \(dv\) to get \(v = e^{r}\).

Apply the integration by parts formula: write \(\int (r^{2} + r + 1) e^{r} \, dr = (r^{2} + r + 1) e^{r} - \int (2r + 1) e^{r} \, dr\), then plan to evaluate the remaining integral using integration by parts again if necessary.

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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.

Choosing u and dv

Selecting which part of the integrand to set as u and which as dv is crucial. Typically, u is chosen as a polynomial or algebraic function that simplifies upon differentiation, while dv is chosen as an easily integrable function like an exponential or trigonometric function.

Integrating Exponential Functions

The integral of an exponential function e^r with respect to r is straightforward, ∫e^r dr = e^r + C. This property is useful when applying integration by parts, as it allows the exponential part to be integrated easily, simplifying the overall integral.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ (r² + r + 1) e^r dr