Problem 210 - Calculus Volume 1

Question

Use basic integration formulas to compute the following antiderivatives. $$ \int \frac{x-1}{x^{2}} d x $$

Step-by-Step Solution

Answer

\( \int \frac{x-1}{x^{2}} \, dx = \ln|x| - \frac{1}{x} + C \).

Step 1: Simplifying the Integrand

Start by simplifying the expression \( \frac{x-1}{x^2} \). This can be rewritten as \( \frac{x}{x^2} - \frac{1}{x^2} \), which simplifies to \( \frac{1}{x} - \frac{1}{x^2} \). This step makes it easier to integrate each term separately.

Step 2: Integrating the First Term

Integrate the first term, \( \int \frac{1}{x} \, dx \). This is a standard integral and its antiderivative is \( \ln|x| + C \), where \( C \) is a constant.

Step 3: Integrating the Second Term

Now, integrate the second term, \( \int -\frac{1}{x^2} \, dx \). This can be rewritten using the power rule for integration. The integral of \( -x^{-2} \) is \( x^{-1} \), which simplifies to \( -\frac{1}{x} \).

Step 4: Combining the Results

Combine the results from Steps 2 and 3. The overall antiderivative is \( \ln|x| + C_1 - \frac{1}{x} + C_2 \). Since integration constants can be combined, the final antiderivative is \( \ln|x| - \frac{1}{x} + C \), where \( C = C_1 + C_2 \).

Key Concepts

Antiderivatives

When we talk about integration in mathematics, we often refer to the process of finding antiderivatives. An antiderivative is essentially a function whose derivative gives us back the original function. Imagine working backwards from a derivative to infer the original function — that's what antiderivation means. It plays a critical role in calculus and is foundational in solving many mathematical problems involving rates of change or areas under curves.

• Think of finding an antiderivative as undoing the process of differentiation.
• Every function has an infinite number of antiderivatives differing by a constant, denoted as "C".
• The constant comes in because the derivative of a constant is zero, meaning it disappears during differentiation.

Understanding antiderivatives helps you grasp the full picture of integral calculus. Integration tells you how to "accumulate" quantities, and antiderivatives show you what was "differentiated" away. It's crucial for students to distinguish between the steps of differentiating and then integrating back, which involves retrieving all aspects of the original function.

Power Rule

The power rule is a basic yet crucial tool in both differentiation and integration. In the context of integration, it helps us convert terms into a format where it’s easy to find antiderivatives. But how does it work?For integration, the power rule states: If you have a term like \( x^n \), its integral is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). This isn't magic—it's just a reverse of what you do when differentiating. The exponential function becomes slightly "fattened" because you're accumulating its rate of change backward.

• By applying the power rule, you get straightforward antiderivatives for any polynomials.
• Remember: increment the exponent by one, then divide by that new value.
• The power rule does not apply when \( n = -1 \) because of a special case—the natural logarithm.

You'll frequently encounter scenarios where powerful rules simplify complex integration tasks. Hopefully, this gives you confidence in knowing which rules to apply and when. It's like having a formula sheet in your head that lets you tackle integrations with ease!

Basic Integration Formulas

Basic integration formulas act as a toolkit for solving a wide array of integral problems quickly and reliably. These formulas are standardized based on common functions you might need to integrate regularly. Let's explore these necessities.Core formulas include:

• The integral of \( x^n \) (where \( n eq -1 \)): \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
• The integral of a constant \( k \): \( \int k \, dx = kx + C \).
• The integral of \( e^x \): \( \int e^x \, dx = e^x + C \).
• The integral of \( \frac{1}{x} \): \( \int \frac{1}{x} \, dx = \ln|x| + C \).

These are just the beginning and build your foundation in integration techniques. Having these formulas at your fingertips enhances your efficiency and ability to approach calculus problems logically and systematically. They are invaluable tools that simplify the process of finding areas, volumes, or solving differential equations.