Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ e^(-3t) sin(4t) dt

검증된 단계별 안내

Recognize that the integral is of the form \(\int e^{at} \sin(bt) \, dt\), where \(a = -3\) and \(b = 4\). This is a standard integral that can be found in the table of integrals.

Recall the formula for the integral: \(\int e^{at} \sin(bt) \, dt = \frac{e^{at}}{a^2 + b^2} (a \sin(bt) - b \cos(bt)) + C\), where \(C\) is the constant of integration.

Substitute the values \(a = -3\) and \(b = 4\) into the formula to express the integral in terms of \(t\).

Write the integral as \(\int e^{-3t} \sin(4t) \, dt = \frac{e^{-3t}}{(-3)^2 + 4^2} (-3 \sin(4t) - 4 \cos(4t)) + C\).

Simplify the denominator and the expression inside the parentheses as much as possible to get the final integral expression.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration of Products of Exponential and Trigonometric Functions

Integrals involving products of exponential and trigonometric functions often require special techniques such as integration by parts or using known integral formulas. Recognizing the form helps in applying the correct formula or method to simplify the integral efficiently.

Use of Integral Tables

Integral tables provide pre-calculated formulas for common integrals, saving time and effort. Knowing how to locate and apply the correct formula from the table is essential, especially for integrals involving combinations of exponential and trigonometric functions.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, useful for integrating products of functions. It involves choosing parts of the integrand as 'u' and 'dv' to simplify the integral, often applied repeatedly for integrals like ∫ e^(at) sin(bt) dt.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ e^(-3t) sin(4t) dt