Evaluate the integrals in Exercises 33–54.
∫(e^(3x) + 5e^(-x)) dx

검증된 단계별 안내

Recognize that the integral is a sum of two separate integrals: \(\int (e^{3x} + 5e^{-x}) \, dx = \int e^{3x} \, dx + \int 5e^{-x} \, dx\).

Recall the integral formula for exponential functions: \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\), where \(a\) is a constant.

Apply the formula to the first integral: \(\int e^{3x} \, dx = \frac{1}{3} e^{3x} + C_1\).

Apply the formula to the second integral, factoring out the constant 5: \(\int 5e^{-x} \, dx = 5 \int e^{-x} \, dx = 5 \left(-e^{-x}\right) + C_2\).

Combine the results of both integrals and include a single constant of integration \(C\): \(\int (e^{3x} + 5e^{-x}) \, dx = \frac{1}{3} e^{3x} - 5 e^{-x} + C\).

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주요 개념

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

Integration of Exponential Functions

Integrating exponential functions involves reversing differentiation rules. For functions like e^(ax), the integral is (1/a)e^(ax) + C, where a is a constant. This rule applies to each term separately in a sum.

Linearity of Integration

Integration is a linear operation, meaning the integral of a sum is the sum of the integrals. This allows us to split ∫(f(x) + g(x)) dx into ∫f(x) dx + ∫g(x) dx, simplifying the evaluation process.

Constant of Integration

When evaluating indefinite integrals, a constant of integration (C) must be added to represent all possible antiderivatives. This accounts for any constant term lost during differentiation.

Evaluate the integrals in Exercises 33–54.∫(e^(3x) + 5e^(-x)) dx