Problem 89 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int 3^{x} d x $$

Step-by-Step Solution

Answer

\( \int 3^x \, dx = \frac{3^x}{\ln(3)} + C \).

Step 1: Recall the Integral Formula for Exponential Functions

The indefinite integral of an exponential function of the form \( a^x \) is given by \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \), where \( a > 0 \) and \( a eq 1 \). Here, \( C \) is the constant of integration.

Step 2: Apply the Formula to the Given Integral

Given the integral \( \int 3^x \, dx \), identify \( a \) as 3. Substitute \( a = 3 \) into the formula: \( \int 3^x \, dx = \frac{3^x}{\ln(3)} + C \).

Step 3: Complete the Solution

Write down the final expression for the indefinite integral using the formula applied to this specific function: \( \int 3^x \, dx = \frac{3^x}{\ln(3)} + C \), where \( C \) is the constant of integration.

Key Concepts

Exponential Functions

Exponential functions are a special class of mathematical functions, often expressed as \( a^x \), where \( a \) is a constant and \( x \) is a variable. These functions are powerful tools in describing growth or decay processes, such as population growth or radioactive decay. They are represented by the equation:

• \( f(x) = a^x \)
• where \( a \) is a real positive constant, and \( a eq 1 \)

The base \( a \) influences the rate of increase or decrease. The larger the base, the faster the exponential function grows. These functions are widely used in various fields, including economics, biology, and physics. Their unique property is that their rate of growth is proportional to their current value, making them suitable for modeling many natural phenomena.

Integration Formula

In calculus, integrating exponential functions involves using a specific integration formula. This formula helps find the antiderivative, which is crucial as it allows us to compute indefinite integrals. For a general exponential function \( a^x \), the integration formula is:

• \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \)
• where \( \ln(a) \) is the natural logarithm of \( a \), and \( C \) is the constant of integration

This formula is applicable as long as \( a > 0 \) and \( a eq 1 \). In our exercise, we use this formula to evaluate the integral of \( 3^x \). We substitute \( a = 3 \) into the formula, resulting in:

• \( \int 3^x \, dx = \frac{3^x}{\ln(3)} + C \)

This outcome provides a general solution for every \( x \). Understanding this formula and how to apply it is essential in solving integrals involving exponential functions.

Constant of Integration

The constant of integration, denoted as \( C \), plays a significant role in the calculation of indefinite integrals. When finding indefinite integrals, you are essentially recovering all possible antiderivatives of a function. Since differentiation erases constant terms, indefinite integrals include \( C \) to account for any constant that might have been present before the function was differentiated.

• \( \int f(x) \, dx = F(x) + C \)
• \( C \) represents an arbitrary constant

The constant ensures completeness of the integral expressions, reflecting all potential solutions. For instance, when calculating \( \int 3^x \, dx \), the expression \( \frac{3^x}{\ln(3)} + C \) includes \( C \) to signify any possible vertical shift in the function. In applied scenarios, the constant may be determined by initial conditions or additional information, tailoring the general solution to specific circumstances.