Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int 4 e^{2 x-1} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function \(4 e^{2x-1}\) is \(2e^{2x-1} + C.\)

Step 1: Identify the Integral Form

The given integral is in the form \(\int a e^{bx+c} dx\), where \(a = 4\), \(b = 2\), and \(c = -1\). So, we will use the formula of exponential integration \(\int a e^{bx} dx = \frac{a}{b} e^{bx} + C\).

Step 2: Apply the Exponential Integration Formula

Apply the formula to our integral: \(\int 4 e^{2x-1} dx = \frac{4}{2}e^{2x-1} + C\). Simplifying gives the integral: \(2e^{2x-1} + C\).

Step 3: Final Solution

Our final answer is the integral of the given function: \(\int 4 e^{2x-1} dx = 2e^{2x-1} + C\). The '+ C' represents the constant of integration, because the integral of a constant is a constant.

Key Concepts

Exponential Integration

Exponential integration involves finding the integral of a function that includes an exponential expression, like \( e^{x} \). This process is fundamental to calculus and is often encountered when dealing with growth and decay problems in real-world scenarios.
Exponential integrals are useful because they allow us to reverse the operation of differentiation on exponential functions.
When dealing with an integral such as \( \int a e^{bx+c} \, dx \), it is important to identify the constants correctly:

• "\( a \)" is the coefficient of the exponential term.
• "\( b \)" and "\( c \)" are parameters inside the exponential function.

For instance, our original integral \( \int 4 e^{2x-1} \, dx \) can be solved by following the rule for exponential integrals. This involves taking \( a = 4 \), \( b = 2 \), and \( c = -1 \), and applying the relevant formula to find the solution.

Integration Formula

The integration formula for exponential functions is a tool that helps simplify the process of finding the antiderivative of expressions involving exponential terms. In this scenario, we employ the formula: \[ \int a e^{bx+c} \, dx = \frac{a}{b} e^{bx+c} + C \]This succinctly provides a method to handle the exponential term efficiently. Breaking this down:

• Divide the coefficient "\( a \)" by the exponent's base coefficient "\( b \)".
• Maintain the exponential expression \( e^{bx+c} \) as is.
• Add \( C \), which stands for the constant of integration, to represent any constant not captured by the antiderivative.

For \( \int 4 e^{2x-1} \, dx \), applying the formula: divide 4 by 2 to get 2, preserving the exponential portion, leading to the simplified integral \( 2e^{2x-1} + C \). Each step reduces complexity, making integration more approachable.

Constant of Integration

When calculating an indefinite integral, it is crucial to include the constant of integration, denoted by \( C \). This ensures that all possible antiderivatives are considered, not just a single one.
When differentiating, any constant term disappears, so when reversing this process through integration, an arbitrary constant \( C \) must be added.
The importance of \( C \) cannot be overstated, as it reflects the entire family of curves represented by the antiderivative. For example, the function \( 2e^{2x-1} \) could intersect the y-axis at various points, depending on the value of \( C \).

• Omitting "\( C \)" can lead to incomplete solutions.
• \( C \) accounts for different scenarios where initial conditions are not given.
• Always including it makes solutions comprehensive and flexible.

So, by adding \( C \) in our solution \( 2e^{2x-1} + C \), we acknowledge the plurality of solutions and maintain mathematical accuracy.