Problem 51 - Applied Calculus

Question

Find the indefinite integrals. $$ \int e^{3 r} d r $$

Step-by-Step Solution

Answer

\( \int e^{3r} \, dr = \frac{1}{3} e^{3r} + C \)

Step 1: Identify the basic integral form

We start by recognizing the integral form we need to use. The integral of an exponential function of the form \( e^{ax} \) is \( \int e^{ax} \; dx = \frac{1}{a} e^{ax} + C \), where \( a \) is a constant and \( C \) is the constant of integration.

Step 2: Determine the constant 'a'

In our integral \( \int e^{3r} \, dr \), the exponent is \( 3r \). Thus, the constant \( a \) in this context is 3.

Step 3: Apply the basic integral formula

Using the formula from Step 1, substitute 3 for \( a \). The integral becomes: \( \int e^{3r} \, dr = \frac{1}{3} e^{3r} + C \).

Step 4: Write the final answer

The indefinite integral of \( e^{3r} \) with respect to \( r \) is \( \frac{1}{3} e^{3r} + C \). This expression includes the constant of integration \( C \), which accounts for any constant that could be part of the original function before differentiation.

Key Concepts

Exponential Functions

Exponential functions are foundational in calculus and many fields of science and engineering. These functions involve a constant base raised to a variable exponent, typically expressed as \( e^x \), where \( e \) is the mathematical constant approximately equal to 2.71828. Exponential functions have the unique property that their rate of growth is proportional to their current value. This means they grow exponentially, making them quite different from polynomial functions.

• The exponential function \( e^x \) is crucial because its derivative with respect to \( x \) is \( e^x \) itself.
• When integrating functions like \( e^{3r} \), you utilize the structure \( e^{ax} \), where \( a \) is a constant, to determine the integral.

Recognizing and understanding exponential functions allow you to work with growth models, population dynamics, and complex systems that exhibit change at an exponential rate.

Constant of Integration

The constant of integration, often denoted by \( C \), plays a significant role in indefinite integrals. When you integrate a function, you obtain a family of functions differing by a constant. This is because differentiation eliminates constants, making their recovery through integration unfeasible without additional information.

• Every indefinite integral comes with this constant because the antiderivative can vary by a constant without affecting the derivative.
• If you integrate \( e^{3r} \), the result \( \frac{1}{3}e^{3r} + C \) signifies that you could add any constant to the function, and it would still have \( e^{3r} \) as its derivative.

The presence of \( C \) highlights that integration is not merely reversing differentiation but finding all possible original functions.

Basic Integration Formula

The integration of exponential functions relies heavily on learning the basic integration formulas. A key formula for indefinite integrals of the form \( \ e^{ax} \) is \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). This formula is central for integrating exponential functions and recognizing patterns related to the exponent.

• When you know \( a \), integrating becomes straightforward by rearranging the integral according to \( \frac{1}{a} \).
• In the example \( \int e^{3r} \, dr \), \( a \) is 3, so the integral becomes \( \frac{1}{3} e^{3r} + C \).

Mastering this formula enables you to efficiently solve integrals involving exponential functions, laying the groundwork for more complex calculus applications.