Problem 19 - Calculus and its applications

Question

Find each integral. $$ \int \frac{10}{x} d x $$

Step-by-Step Solution

Answer

The integral is \( 10 \ln |x| + C \).

Step 1: Identify the form of the integral

We are given the integral \( \int \frac{10}{x} \, dx \). This integral is in the form \( \int \frac{a}{x} \, dx \), where \( a = 10 \). This follows the standard form for the integral of \( \frac{1}{x} \), which is a natural logarithmic function.

Step 2: Apply the standard integral formula

The integral \( \int \frac{a}{x} \, dx \) can be solved using the formula \( a \ln |x| + C \), where \( C \) is the constant of integration. Substituting \( a = 10 \) into this formula, we get \( 10 \ln |x| + C \).

Step 3: Write the final solution

The solution to the integral \( \int \frac{10}{x} \, dx \) is thus \( 10 \ln |x| + C \), where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

In the world of integral calculus, integration techniques are essential tools. They help us solve various integral problems. When we look at the integral of the form \( \int \frac{10}{x} \, dx \), we can recognize it as a basic type of problem, which is often the starting point for learners.

The technique used here is straightforward but powerful. It leverages the formula for integrating \( \frac{1}{x} \).

This is a common integral and one of the first that students encounter. Key things to look for in such problems include:

• Identifying constants: Here, \( a = 10 \) is the constant in \( \frac{a}{x} \).
• Recognizing the natural logarithmic form: the integral \( \int \frac{1}{x} \, dx \) results in a logarithmic function.
• Applying known formulas efficiently to simplify calculations.

Remember, mastering these basics allows you to tackle more complex integrals with confidence later on.

Natural Logarithm

The natural logarithm, often written as \( \ln \), is a logarithm with a base of \( e \), where \( e \) is an irrational constant approximately equal to 2.718. In integral calculus, the natural logarithm frequently appears when dealing with integrals of the form \( \int \frac{1}{x} \, dx \).

Integrating such functions leads to \( \ln |x| \).

This relationship is important because:

• The derivative of \( \ln |x| \) is \( \frac{1}{x} \), which shows the reverse operation in calculus (differentiation).
• Logarithms offer a way to simplify problems involving exponentials. This aligns well with exponential growth and decay problems you might encounter in the real world.
• \( \ln |x| \) specifically accounts for both positive and negative \( x \), thanks to the absolute value signs.

Understanding how natural logarithms function within integration provides a deeper insight into solving and simplifying math problems efficiently.

Constant of Integration

When we find an indefinite integral, we frequently see the term "constant of integration," denoted as \( C \). This concept is crucial because when you integrate, you are essentially performing the reverse of differentiation.

Since differentiation loses any constant (because the derivative of any constant is zero), the integration process needs to account for this "lost" constant.
Therefore, we always add \( C \) to indefinite integrals, like in \( 10 \ln |x| + C \) for our problem.

• \( C \) represents any constant value, as multiple solutions exist if the derivative leads back to the given function.
• This constant is crucial in real-life applications, setting initial conditions or specific solution parameters.

So, it’s important to always include it when working with indefinite integrals to ensure the solution set remains complete.