Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt

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Identify the parts of the integral for integration by parts. Let \( u = t^2 \) (a polynomial function) and \( dv = e^{4t} dt \) (an exponential function).

Compute the derivatives and integrals needed: find \( du = \frac{d}{dt}(t^2) dt = 2t dt \) and \( v = \int e^{4t} dt = \frac{1}{4} e^{4t} \).

Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions to get \( \int t^2 e^{4t} dt = t^2 \cdot \frac{1}{4} e^{4t} - \int \frac{1}{4} e^{4t} \cdot 2t dt \).

Simplify the integral: \( \int t^2 e^{4t} dt = \frac{t^2}{4} e^{4t} - \frac{1}{2} \int t e^{4t} dt \). Notice that the remaining integral \( \int t e^{4t} dt \) also requires integration by parts.

Repeat integration by parts on \( \int t e^{4t} dt \) by letting \( u = t \) and \( dv = e^{4t} dt \), then substitute back to express the original integral fully in terms of elementary functions and integrals.

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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.

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, and dv as a function that is easy to integrate. For example, polynomial terms often serve well as u.

Repeated Application of Integration by Parts

Some integrals, like ∫ t² e^(4t) dt, require applying integration by parts multiple times. Each application reduces the power of the polynomial until the integral becomes straightforward. Keeping track of terms carefully ensures correct evaluation.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ t² e^(4t) dt