Problem 47 - Thomas Calculus

Question

Evaluate the integrals. \(\int 5^{x} d x\)

Step-by-Step Solution

Answer

The integral is \(\int 5^x \, dx = \frac{5^x}{\ln 5} + C\).

Step 1: Identify the Type of Integral

This integral involves an exponential function, specifically in the form of base 5 raised to a variable exponent. Therefore, we will use the formula for integrating exponential functions of the form \(\int a^{x} \, dx = \frac{a^{x}}{\ln a} + C\).

Step 2: Apply the Exponential Integral Formula

Using the formula from Step 1, substitute \(a = 5\) into \(\int 5^{x} \, dx = \frac{5^{x}}{\ln 5} + C\), where \(C\) is the constant of integration.

Key Concepts

Exponential Function

An exponential function is a powerful mathematical concept that occurs throughout various mathematical and real-world applications. The function typically takes the form \(a^x\), where \(a\) is a constant base and \(x\) is the exponent. In this type of function:

• The base \(a\) is a positive constant not equal to 1.
• The function grows or decays rapidly due to the power of the exponent \(x\).
• If \(a > 1\), the function exhibits exponential growth.
• If \(0 < a < 1\), the function shows exponential decay.

Exponential functions are key in fields such as biology for modeling population growth, finance for compound interest calculations, and physics for radioactive decay analysis. Integrating exponential functions requires a specific approach since their derivatives are proportional to the function itself.

Integration Techniques

Integration techniques are mathematical strategies used to solve integrals, which are operations to find the area under curves represented by functions. When dealing with exponential functions, like \(5^x\), a specific technique for integration is employed. This involves using a known formula:

• For exponential functions in the form of \(a^x\), the integral is \(\int a^x \, dx = \frac{a^x}{\ln a} + C\).
• The constant \(a\) gets raised to the power of \(x\), with the result divided by the natural logarithm of \(a\).
• This formula takes into account the unique properties of exponential growth and decay.

Integration can seem complex, but by applying these strategies step-by-step, you can navigate through the problem to find a solution effectively. This technique is one of many that simplifies the process of finding integrals, especially in calculus problems involving growth and decay.

Constant of Integration

The constant of integration, denoted as \(C\), plays a crucial role in indefinite integrals. While definite integrals provide a specific value, indefinite integrals yield a broad family of functions.

• Every integral requires adding a constant of integration because the process of differentiation eliminates constants; we must account for any constant that could have existed.
• This constant represents the idea that there could be several functions that differentiate to the same derivative.
• If you take the derivative of any of these functions, the result will be the same, but their integral representations slightly differ by a constant value.

Remember, the constant \(C\) becomes essential in equations and modeling because it accounts for initial conditions or offsets, ensuring the solved integral accurately reflects the starting scenario or benchmark in specific problems. Thus, it completes the integration process by encapsulating all possible original functions.