Evaluate the integrals in Exercises 87–96.
87. ∫ 5ˣ dx

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Recognize that the integral involves an exponential function with base 5, which can be written as \$5^x$.

Recall the general formula for integrating an exponential function with base $a$: $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$, where $a > 0$ and $a \neq 1$.

Apply this formula to the integral $\int 5^x \, dx$, substituting $a = 5$.

Write the integral as $\int 5^x \, dx = \frac{5^x}{\ln(5)} + C$, where $C$ is the constant of integration.

This expression represents the antiderivative of \$5^x$.

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주요 개념

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

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. Understanding how these functions behave and their properties is essential for integrating expressions like 5^x.

Integration of Exponential Functions

The integral of an exponential function a^x with respect to x is (a^x) / ln(a) + C, where ln(a) is the natural logarithm of the base. This formula is crucial for solving integrals involving exponential terms.

Natural Logarithm (ln)

The natural logarithm ln(x) is the inverse of the exponential function e^x. It appears in the integration formula of exponential functions and is important for simplifying and understanding the results of such integrals.

Evaluate the integrals in Exercises 87–96.87. ∫ 5ˣ dx