Problem 58 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int 2 e^{3 x} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{2}{3} e^{3x} + C \).

Step 1: Identify the Integral Format

The integral we need to compute is \( \int 2 e^{3x} \, dx \). This matches the format \( \int a e^{bx} \, dx \) where \( a = 2 \) and \( b = 3 \). We will use the rule for integrating the exponential function.

Step 2: Apply the Integration Rule for Exponential Functions

The rule for integrating \( e^{bx} \) is \( \int e^{bx} \, dx = \frac{1}{b} e^{bx} + C \). We will apply this rule to \( 2 e^{3x} \), making sure to include the constant multiple 2.

Step 3: Integrate each Component

First, factor out the constant 2, then integrate: \( \int 2 e^{3x} \, dx = 2 \int e^{3x} \, dx \). Using the integration rule, we get \( 2 \times \frac{1}{3} e^{3x} = \frac{2}{3} e^{3x} \).

Step 4: Add the Constant of Integration

Since this is an indefinite integral, always add a constant of integration \( C \). Thus, the integral becomes \( \frac{2}{3} e^{3x} + C \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. These functions are written in the general form \( e^{bx} \) where \( e \) is the base of the natural logarithm (approximately 2.71828) and \( bx \) is the exponent. In our exercise, we have the function \( e^{3x} \). Here, the base \( e \) is raised to the power of \( 3x \). This type of function is very special in calculus

• It grows very rapidly or decays depending on the sign of the exponent.
• The derivative of \( e^{bx} \) is \( be^{bx} \), making it unique.

This property makes exponential functions particularly important in many areas such as population growth, radioactive decay, and finance. In the context of calculus, they are often found in problems involving continuous growth or decay processes, making them crucial in many applications.

Integration by Parts

To solve integrals involving multiplicative relationships, sometimes the method of integration by parts is needed. This is often used when simple application of integration rules won't suffice. The integration by parts formula is derived from the product rule for differentiation

• Given two functions \( u \) and \( v \), it states: \( \int u \, dv = uv - \int v \, du \).

However, in our integral, \( \int 2 e^{3x} \, dx \), we don't actually need to use integration by parts. This integral can be solved more directly using a substitution rule. We identify that it's easier to integrate the exponential directly using the formula \( \int e^{bx} \, dx = \frac{1}{b} e^{bx} + C \). Understanding when and when not to use integration by parts saves time and simplifies solving integrations, especially for exponential functions where a direct approach often suffices.

Constant of Integration

When calculating indefinite integrals, the constant of integration \( C \) is a key element. It appears because integration is essentially the reverse operation of differentiation. When you differentiate a constant, the result is zero, making it "disappear" from equations. Therefore, when providing the antiderivative, we add \( C \) to account for any constant that might have been in the original function.

• For example, in the integral \( \int 2 e^{3x} \, dx \), you end up with \( \frac{2}{3} e^{3x} + C \).
• Adding \( C \) ensures the family of all possible solutions is represented.

This is why it’s important to remember that indefinite integrals always include a constant of integration. It represents all possible vertical shifts of the antiderivative, maintaining the integrity and completeness of the solution.