Question

In the following exercises, compute each indefinite integral. $$ \int 2^{x} d x $$

Short answer

\( \int 2^x \, dx = \frac{2^x}{\ln(2)} + C \).

Step-by-step solution

Identify Integral Type

The integral \( \int 2^x \, dx \) is an exponential integral, which involves an exponential function with a base other than \( e \).

Recall Integral Formula

The formula to integrate \( a^x \), where \( a \) is a constant, is \( \frac{a^x}{\ln(a)} + C \), where \( C \) is the constant of integration.

Apply the Formula

Use the formula from Step 2 to evaluate the integral: \( \int 2^x \, dx = \frac{2^x}{\ln(2)} + C \).

Write Final Answer

The indefinite integral evaluated is \( \frac{2^x}{\ln(2)} + C \), where \( C \) is the constant of integration.

Key concepts

Exponential Functions

Exponential functions are a type of mathematical function where the variable is in the exponent. For instance, in the function \( f(x) = a^x \), \( a \) is a constant known as the base while \( x \) is the exponent.The most common base is the mathematical constant \( e \), approximately equal to 2.718. This is known as the natural exponential function. However, in some cases, the base can be numbers other than \( e \), like 2, 3, or even a fraction like \( \frac{1}{2} \).Exponential functions have unique properties:

• They grow (or decay) at a rate proportional to their current value.
• They are always positive for all real-number values of \( x \).

Understanding exponential functions is crucial as they model real-world scenarios such as population growth, radioactive decay, and compounding interest. In calculus, these functions frequently appear in problems related to rates of change and growth models.

Integration Techniques

Integration, the process of finding integrals, is a core concept in calculus. It helps in finding the area under a curve and the accumulation of quantities. When integrating exponential functions like \(2^x\), specific integration techniques and formulas need to be applied.In the case of exponential functions with a non-\( e \) base, such as \( 2^x \), the formula \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \) is used:

• \( a \) is the base of the exponential function.
• \( \ln(a) \) represents the natural logarithm of the base \( a \).

For the given problem \( \int 2^x \, dx \), applying this formula allows us to calculate the integral properly, ensuring we capture the behavior of the function as it updates along its interval. Thus, integration using these formulas helps unlock valuable insights for exponential functions, aiding in solving broader mathematical problems.

Constant of Integration

In indefinite integrals, such as \( \int 2^x \, dx \), adding a constant of integration, denoted as \( C \), is important. This is because the process of integration is essentially the reverse of differentiation, and it reflects the fact that the original function could have been shifted vertically by any constant value.The constant of integration accounts for:

• The infinite set of antiderivatives that a function could have, all differing by a constant.
• The possibility that there might not be enough information to determine a specific starting value for the function.

When you compute an indefinite integral, like \( \frac{2^x}{\ln(2)} + C \), it represents all potential functions that could have this derivative, showcasing how flexible integration can be. Thus, always remember to include \( C \) when calculating indefinite integrals to accurately capture the full set of possible solutions.