Evaluate ∫ 4ˣ dx.

Verified step by step guidance

1. Recognize that the integral involves an exponential function with base 4, which can be written as \$4^x$.
2. 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 \(a \neq 1\).
3. Apply this formula to the integral \(\int 4^x \, dx\), identifying \(a = 4\).
4. Write the integral as \(\int 4^x \, dx = \frac{4^x}{\ln(4)} + C\), where \(C\) is the constant of integration.
5. Conclude that the integral is expressed in terms of the exponential function divided by the natural logarithm of the base, plus the constant of integration.

Key Concepts

Exponential Functions

An exponential function has the form a^x, where the base a is a positive constant. Understanding how these functions behave and their properties is essential for integrating expressions like 4^x.

Integration of Exponential Functions

The integral of a^x with respect to x is (a^x) / ln(a) + C, where a > 0 and a ≠ 1. This formula is derived using substitution and the natural logarithm, and it is key to 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 for a^x because the derivative of a^x involves ln(a), making it crucial for correctly evaluating the integral.