Problem 53 - Applied Calculus

Question

Find the indefinite integrals. $$\int\left(1+\frac{1}{p}\right) d p$$

Step-by-Step Solution

Answer

The indefinite integral is \( p + \ln|p| + C \).

Step 1: Separate the Terms

Start by separating the integral into two distinct parts, which simplifies the integration process: \[ \int \left(1 + \frac{1}{p} \right) dp = \int 1 \, dp + \int \frac{1}{p} \, dp \] This allows us to handle each term independently.

Step 2: Integrate the Constant Term

Integrate the constant \(1\) with respect to \(p\). The integral of a constant \(c\) is \(cp\) plus a constant of integration. Therefore: \[ \int 1 \, dp = p \]

Step 3: Integrate the Reciprocal Term

Now, integrate \( \frac{1}{p} \) with respect to \(p\). The integral of \( \frac{1}{p} \) is the natural logarithm of the absolute value of \(p\): \[ \int \frac{1}{p} \, dp = \ln |p| \]

Step 4: Combine the Results and Add the Constant of Integration

Combine the results from Step 2 and Step 3 to form the general solution, adding a constant of integration \(C\): \[ \int \left(1 + \frac{1}{p} \right) dp = p + \ln |p| + C \] This represents the indefinite integral of the given function.

Key Concepts

Integration Techniques

Integration is a key concept in calculus, helping us find the area under the curve of a function. There are several techniques to solve integrals. When faced with an integral, it’s crucial to recognize which technique to apply for optimal results.

For the given exercise, separating the terms into simpler components is an effective integration technique. This involves breaking down the integral into simpler parts that are easier to solve individually. In this case:

• The integral of a constant term.
• The integral of a fraction, particularly \( \frac{1}{p} \).

This separation simplifies the process significantly, allowing us to handle each term one at a time.

Once separated, we can apply basic integration rules to each part. This is often one of the first steps in solving integrals, especially when dealing with multiple terms.

Natural Logarithm

The natural logarithm is an important function in mathematics, often denoted as \( \ln \). It has a base of \( e \), which is approximately 2.718.

• In calculus, the natural logarithm appears frequently, especially in integration.
• It's the antiderivative of \( \frac{1}{p} \).

In the current exercise, we integrated the term \( \frac{1}{p} \) with respect to \( p \). The result is \( \ln |p| \). The absolute value bars in \( \ln |p| \) ensure the argument of the logarithm is positive, which is necessary since the logarithm is only defined for positive numbers. Thus, the natural logarithm not only helps in integration but also ensures the solution remains within the bounds of mathematical rules.

Understanding the properties of natural logarithms can greatly aid in recognizing potential simplifications during integration.

Constant of Integration

The constant of integration, denoted as \( C \), is vital when dealing with indefinite integrals. Unlike definite integrals, which yield specific numbers, indefinite integrals produce a family of functions.

• This family is defined by \( C \), an arbitrary constant.
• It ensures that all possible antiderivatives are represented.

When integrating \( \integ (1 + \frac{1}{p} eg ) dp eg \), we find the general solution \( p + \ln |p| + C \). Here, the \( C \) accounts for any constant term that might differentiate one antiderivative from another.

It's important to always remember to include \( C \) when writing the solution to an indefinite integral, as it signifies the additive nature of the integration process.