Evaluate the integrals in Exercises 33–54.
∫8e^(x+1) dx

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Recognize that the integral is of the form $\int 8 e^{x+1} \, dx$, where the integrand is an exponential function with a linear argument in the exponent.

Use the property of exponents to rewrite the integrand as \$8 e^{x} e^{1}$, or equivalently factor out the constant $e^{1}$ since it does not depend on $x$.

Rewrite the integral as $8 e^{1} \int e^{x} \, dx$, separating the constant multiplier from the integral.

Recall the integral formula for the exponential function: $\int e^{x} \, dx = e^{x} + C$, where $C$ is the constant of integration.

Apply the integral formula and multiply back by the constants to express the antiderivative as $8 e^{1} e^{x} + C$, which can be combined as $8 e^{x+1} + C$.

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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. For functions like e^(ax+b), the integral is (1/a)e^(ax+b) + C, where a and b are constants. Recognizing the inner function helps apply the correct formula.

Constant Multiple Rule

The constant multiple rule states that a constant factor can be pulled out of the integral. For example, ∫k*f(x) dx = k*∫f(x) dx. This simplifies integration by isolating constants from variable expressions.

Indefinite Integrals and Integration Constants

Indefinite integrals represent families of functions differing by a constant, denoted as + C. This constant accounts for all possible antiderivatives since differentiation of a constant is zero.

Evaluate the integrals in Exercises 33–54.∫8e^(x+1) dx