Problem 90 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int 2^{x} d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral Type

Recognize that the given integral is of the form \( \int a^x \, dx \), where \( a = 2 \). This is an exponential function with base 2.

Step 2: Apply the Exponential Integral Formula

Recall that for an exponential function \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \), where \( a \) is a positive constant and \( C \) is the constant of integration.

Step 3: Substitute and Simplify

Substitute \( a = 2 \) into the formula to get: \( \int 2^x \, dx = \frac{2^x}{\ln(2)} + C \). This is the solution to the integral.

Key Concepts

Exponential Integral

An exponential integral involves integrating a function of the form \( a^x \), where \( a \) is a constant and \( x \) is the variable of integration. This type of integral is fundamental in calculus since it provides a method to solve problems involving growth and decay. Exponential integrals capture scenarios where quantities change rapidly, as seen in applications like population dynamics and radioactive decay. The key to solving exponential integrals is recognizing the base of the exponential function and applying the correct formula. For integration, the formula for solving \( \int a^x \, dx \) is \( \frac{a^x}{\ln(a)} + C \). This formula shows how the base \( a \) and its natural logarithm, \( \ln(a) \), play roles in the solution.
There are a few steps to efficiently solve exponential integrals:

• Identify the base \( a \).
• Use the corresponding formula \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \).
• Substitute \( a \) with the actual value given in the problem, as done here with \( a = 2 \).
• Remember that for the exponential integral to be defined, \( a > 0 \) and \( a eq 1 \).

Understanding how to integrate exponential functions opens the door to tackling diverse mathematical and real-world applications.

Constant of Integration

The constant of integration, denoted as \( C \), is an essential part of indefinite integrals. When solving indefinite integrals, you find a family of functions that differ only by a constant. These constants arise because integration is the inverse operation of derivation, and the derivative of a constant is zero. Hence, when you integrate and do not know the original constant that was derived away, you represent it with \( C \).
Think of the constant of integration as:

• A placeholder for any fixed value that makes the antiderivative unique for specific conditions.
• Important when appending initial conditions or boundary conditions in physics and engineering problems.
• Indicating that all indefinitely integrated functions possess a family of curves, each shifted vertically by \( C \).

In the example \( \int 2^x \, dx = \frac{2^x}{\ln(2)} + C \), every possible antiderivative graph is represented, showcasing how the constant of integration \( C \) completes the family of integral solutions.

Integral Calculus

Integral calculus is an essential branch of mathematics closely tied to differential calculus. While differential calculus focuses on rates of change and slopes, integral calculus is concerned with accumulation and areas under curves. Indefinite integrals, like the one solved here, find antiderivatives of functions. These antiderivatives represent a broad family of curves, differing only by a constant. Understanding integral calculus involves grasping a range of concepts:

• Antiderivatives: The inverse of finding the derivative, which provides a function whose derivative matches the integrand.
• Indefinite Integrals: Concerned with finding the general form of antiderivatives, symbolically written without bounds as \( \int f(x) \, dx \).
• Fundamental Theorem of Calculus: Links differentiation and integration, meaning any function can be reconstructed through its derivative and indefinite integral.
• Applications: Helps in calculating areas, volumes, central points, and solutions to differential equations across diverse fields.

Integral calculus plays a crucial role in scientific inquiry and engineering analysis. Grasping its principles allows for insights into the patterns and processes of dynamic systems.