Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx

검증된 단계별 안내

Identify the integral to solve: \(\int x^{5} e^{3x} \, dx\).

Recognize that this integral involves a product of a polynomial \(x^{5}\) and an exponential function \(e^{3x}\), which suggests using integration by parts.

Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Choose \(u = x^{5}\) (since its derivative simplifies) and \(dv = e^{3x} \, dx\) (since it integrates easily).

Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 5x^{4} \, dx\), and integrate \(dv\) to get \(v = \frac{1}{3} e^{3x}\).

Apply the integration by parts formula: \(\int x^{5} e^{3x} \, dx = x^{5} \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 5x^{4} \, dx\). This reduces the original integral to a similar integral with a lower power of \(x\), which you can solve by repeating integration by parts until the polynomial term is eliminated.

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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. This method is especially useful when integrating products like x⁵ and e³ˣ.

Choosing u and dv

Selecting which part of the integrand to assign as u and which as dv is crucial for simplifying the integral. Typically, u is chosen as a polynomial (like x⁵) because its derivative simplifies, and dv as an exponential or trigonometric function, which is easy to integrate.

Repeated Application of Integration by Parts

When the polynomial degree is high, integration by parts may need to be applied multiple times, reducing the power of x step-by-step. This iterative process continues until the polynomial term is eliminated, allowing the integral to be evaluated fully.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ x⁵ e³ˣ dx