Problem 34 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{1}{x}\left(1-x e^{x}\right) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(\ln |x| - e^x + C\).

Step 1: Break Down the Integral

Let's first express the given integral:\[\int \frac{1}{x}(1 - x e^x) \, dx\]This can be separated into two simpler integrals:\[\int \frac{1}{x} \, dx - \int e^x \, dx\]Let's solve these integrals separately.

Step 2: Solve the First Integral

The first integral is:\[\int \frac{1}{x} \, dx\]This is a basic integral that results in the natural logarithm:\[\ln |x| + C_1\]

Step 3: Solve the Second Integral

The second integral is:\[\int e^x \, dx\]The integral of \(e^x\) is simply:\[e^x + C_2\]

Step 4: Combine the Results

Now, combine the results from Step 2 and Step 3 and don't forget to subtract the second result from the first:\[\ln |x| - e^x + C\]where \(C\) is the constant of integration, which incorporates both \(C_1\) and \(C_2\).

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, and several techniques can help solve a wide range of integrals. In this problem, we need to find the indefinite integral of \[\int \frac{1}{x}(1 - x e^x) \, dx.\]We can simplify the integral by breaking it into smaller parts that are easier to manage. This method is called **integration by decomposition**. It allows us to work on each part separately, making the process more straightforward. For this specific integral, we decompose it into:

• The integral of the first part: \(\int \frac{1}{x} \, dx\)
• The integral of the second part: \(\int x e^x \, dx\)

Once decomposed, you can apply known integration rules to tackle each part:- **Polynomial and rational functions**: Often involve basic rules or might require small algebraic manipulations.- **Exponential functions**: Have straightforward integration rules that use the natural base \(e\).Breaking down integrals simplifies the problem, like solving a puzzle by dividing the task into smaller pieces. This technique can be extended to more complex integrals by combining different strategies, such as substitution or integration by parts.

Natural Logarithm

The natural logarithm, denoted by \(\ln x\), is the inverse function of the natural exponential function \(e^x\). It's a vital tool in calculus, especially when dealing with integrals involving exponential growth and decay.In the exercise, one part of the integral involves \(\int \frac{1}{x} \, dx\). When integrated, this yields \(\ln|x|\), a result derived directly from the properties of the natural logarithm.The formula \( \int \frac{1}{x} \, dx = \ln|x| + C\) is a standard result you'll often use when facing integrals with this pattern. It's crucial to remember:

• This result only applies when \(x eq 0\).
• The absolute value in \(\ln |x|\) ensures the logarithm remains defined for negative values of \(x\).
• The constant \(C\) represents all possible shifted antiderivatives since the integral is indefinite.

Understanding the natural logarithm's relationship with its derivative and its antiderivative is essential. Recognizing the connection between \(\frac{1}{x}\) and \(\ln|x|\) will help in various integration problems.

Exponential Function

The exponential function \(e^x\) is unique due to its property where its rate of change is proportional to its value. This characteristic simplifies finding its antiderivative since the exponential function integrates to itself:\[\int e^x \, dx = e^x + C\]In the context of our exercise, the second part deals with:\[\int e^x \, dx\]which directly results in \(e^x\) after integration.Here are key points to note about the exponential function when integrating:

• It's one of the few functions that remain unchanged through differentiation and integration.
• Adding a constant \(C\) ensures that all potential antiderivatives are represented, covering function shifts vertically.
• In combination with other components like coefficients or variables, we might use additional techniques.

The exponential function's simplicity often makes it an anchor point in calculus when expressing solutions to growth or decay problems. Knowing how to integrate \(e^x\) gives you a solid base for tackling complex integrals that include an exponential component.