Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) dx

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Identify the parts of the integral for integration by parts. Let \( u = x \) (a polynomial function) and \( dv = e^{3x} dx \) (an exponential function).

Compute the derivatives and integrals needed: find \( du = dx \) by differentiating \( u \), and find \( v = \int e^{3x} dx \). To integrate \( e^{3x} \), recall that \( \int e^{ax} dx = \frac{1}{a} e^{ax} + C \).

Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions for \( u, v, du \) into this formula.

Set up the resulting integral after substitution, which should be simpler to evaluate than the original integral. This will involve integrating an exponential function.

Evaluate the remaining integral and combine all terms. Don't forget to add the constant of integration \( + C \) at the end.

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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 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 integration process.

Exponential Functions

Exponential functions have the form e^(kx), where k is a constant. Their derivatives and integrals are proportional to the original function, making them straightforward to handle in integration, especially when combined with polynomial terms.

Choosing u and dv in Integration by Parts

Selecting u and dv correctly is crucial for simplifying the integral. Typically, u is chosen as a polynomial or algebraic function that simplifies upon differentiation, while dv is chosen as the remaining part that is easy to integrate, such as an exponential function.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫x e^(3x) dx