Problem 87 - Thomas Calculus

Question

Evaluate the integrals. \begin{equation}\int 5^{x} d x\end{equation}

Step-by-Step Solution

Answer

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

Step 1: Identify the integral to evaluate

We are given the integral \( \int 5^x \, dx \). Our task is to find the indefinite integral of this function.

Step 2: Recall the integral rule for exponential functions

When integrating an exponential function of the form \( b^x \), where \( b \) is a constant, the indefinite integral formula is: \[ \int b^x \, dx = \frac{b^x}{\ln(b)} + C \] where \( C \) is the constant of integration.

Step 3: Apply the integration formula

In the given integral, \( b = 5 \). Applying the formula, we get: \[ \int 5^x \, dx = \frac{5^x}{\ln(5)} + C \]

Step 4: Simplify and write the final answer

There are no further simplifications needed, so the answer to the integral is: \[ \frac{5^x}{\ln(5)} + C \] Make sure to include \( C \), the constant of integration, in the final answer.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that involves exponents. They are expressed in the form of \( b^x \), where \( b \) is a constant base and \( x \) is the exponent, which is usually a variable. These functions are fundamental in many areas of mathematics because they describe growth and decay processes, such as population growth and radioactive decay.

One of the key characteristics of exponential functions is that their rate of change is proportional to their current value. This property makes them unique in comparison to polynomial functions.

• If the base \( b \) is greater than one, the function \( b^x \) will represent exponential growth.
• If \( b \) is between zero and one, the function \( b^x \) indicates exponential decay.

When dealing with integrals involving exponential functions, recognizing this growth or decay behavior helps in understanding how the integral behaves as well.

Indefinite Integral

The indefinite integral, also known as an antiderivative, is a fundamental concept in calculus. It represents the reverse operation of differentiation and is used to find the original function given its derivative. The indefinite integral of a function \( f(x) \) is denoted as \( \int f(x) \, dx \).

In this context, the integral does not have specific limits. Hence, it results in a family of functions rather than a single value.

• The indefinite integral of a function \( f(x) \) is the set of all antiderivatives of \( f \).
• Unlike definite integrals, which compute the area under the curve over a defined interval, indefinite integrals offer a general solution without bounds.

Integrals involving exponential functions require special rules, such as the one used in our exercise: \( \int b^x \, dx = \frac{b^x}{\ln(b)} + C \). This reflects the specific way exponential functions grow or decay.

Constant of Integration

The constant of integration is a crucial component when solving indefinite integrals. It represents the constant that arises due to the antiderivative process, where differentiating this constant results in zero. Thus, it does not affect the differentiation of the function.

When integrating, this constant ensures that all possible antiderivatives are considered. Since the process of taking the derivative of a constant results in zero, multiple functions can have the same derivative.

• The constant of integration is usually denoted by \( C \).
• Without including \( C \) in the solution, the set of potential solutions would be incomplete.

For instance, in the integral \( \int 5^x \, dx = \frac{5^x}{\ln(5)} + C \), \( C \) takes into account any vertical shift in the original function that does not change its derivative.