Problem 6 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{3 e^{-x}+2 e^{3 x}}{e^{2 x}}\right) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \[ -e^{-3x} + 2e^{x} + C.\]

Step 1: Simplify the Integrand

Rewrite the expression inside the integral by splitting the fraction: \[ \frac{3 e^{-x}+2 e^{3 x}}{e^{2x}} = \frac{3 e^{-x}}{e^{2x}} + \frac{2 e^{3x}}{e^{2x}} \] which simplifies to \[ 3 e^{-3x} + 2 e^{x}. \]

Step 2: Write the Indefinite Integral

Now, write the integral in simplified form: \[ \begin{aligned} \ \ \ \ \ \ \ \ omega \ \ \int\left(3 e^{-3x}+2 e^{x}\right)dx. \end{aligned} \]

Step 3: Integrate Each Term Separately

Find the antiderivative of each term: For the first term, we have \[ \ \ \ \ \ omega \ \[ \int 3 e^{-3x} \mathrm{d}x = 3 \frac{e^{-3x}}{-3} = -e^{-3x} \ omega \] while for the second term, we have \[ \int 2 e^{x} \text{d}x = 2 e^{x}.\tex\] \ omega \]

Step 4: Combine the Results

Add the antiderivatives of each term: \[ -e^{-3x} + 2e^{x} + C.\]

Key Concepts

Integral Calculus

Integral calculus is one of the two main branches of calculus, the other being differential calculus. While differential calculus focuses on rates of change, integral calculus is all about accumulation of quantities and the areas under and between curves.

In simple terms, integrals help us calculate the total size or value, such as the area under a curve or the total distance traveled by an object. Indefinite integrals, specifically, are used to find the so-called antiderivative of a function. This means reversing the process of differentiation.

In this exercise, we are dealing with indefinite integrals. The goal is to find the general function whose derivative is the given function inside the integral.

Exponential Functions

Exponential functions are a type of functions where the variable is in the exponent. The general form of an exponential function is \( f(x) = a \cdot b^x\) where \(a\) and \(b\) are constants, and \(x\) is the variable.

Exponential functions have many applications, from modeling population growth to calculating compound interest. They are unique because their rate of increase or decrease is proportional to their current value.

In our integral problem, we encounter exponential functions like \(e^{-x}\), \(e^{3x}\), and \(e^{2x}\). A neat property of the natural exponential function \(e^x\) is that its derivative and its integral are quite straightforward: \( \frac{d}{dx} e^x = e^x \) and \( \int e^x dx = e^x + C \).

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. Finding the antiderivative is the reverse process of differentiation.

When we find an antiderivative, we also include a constant of integration, usually denoted as \(C\), because the derivative of a constant is zero. Hence, there are infinitely many antiderivatives for any given function, differing by a constant.

In our provided solution, we performed separate integrations for each term:

• For \(3 e^{-3x}\), the antiderivative is \int 3 e^{-3x} dx = -e^{-3x}\.
• For \(2 e^x\), we find \int 2 e^x dx = 2e^x\.

Combining these, we get the general solution: \(-e^{-3x} + 2e^x + C\).