Problem 18 - Calculus and its applications

Question

Find each integral. $$ \int \frac{d x}{x^{2}} $$

Step-by-Step Solution

Answer

The integral is \( -\frac{1}{x} + C \).

Step 1: Identify the Integral Type

We are given the integral \( \int \frac{dx}{x^2} \). This is a simple power of \( x \) in the denominator. We can express this integral in terms of a power function: \( \int x^{-2} \, dx \).

Step 2: Apply the Power Rule for Integration

The formula for integrating a function of the form \( x^n \) is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \). In this case, \( n = -2 \), so our integral becomes \( \int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C \).

Step 3: Simplify the Expression

Now, let's simplify the expression \( \frac{x^{-1}}{-1} + C \). This can be rewritten as \( -x^{-1} + C \). Since \( x^{-1} = \frac{1}{x} \), the expression becomes \( -\frac{1}{x} + C \).

Step 4: Write the Final Answer

We have determined the integral of \( \int \frac{dx}{x^2} \) is \( -\frac{1}{x} + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a key technique used for finding antiderivatives of power functions. It states that the integral of a function of the form \( x^n \), where \( n eq -1 \), can be resolved using the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

The formula essentially "increases" the power by 1 and divides by the new power. It's the reverse process of differentiation.
In our exercise, we start by recognizing \( \int \frac{dx}{x^2} \) as \( \int x^{-2} \, dx \). By applying the Power Rule, we transform the problem into finding the new exponent and coefficient, excluding the exception of \( n = -1 \), where a different approach (natural logarithm) is needed. The calculation is relatively straightforward: raise the exponent by one, and divide by this new exponent. This gives us \( \frac{x^{-1}}{-1} \), which simplifies to the solution.

Constant of Integration

The constant of integration, represented as \( C \), plays a crucial role in indefinite integrals. When integrating, you are essentially reversing the process of differentiation, which means various functions can share the same derivative. Adding \( C \) acknowledges this family of functions.
Imagine finding the derivative of a linear function such as \( x + 3 \) and \( x + 7 \). Both will yield the same derivative: 1. Thus, their antiderivatives can only be distinct with a constant added. This constant \( C \) captures all potential vertical shifts of a function's graph.

• It's not a fixed number.
• Represents any possible constant that could be added to the antiderivative.
• Maintains the generality of solutions.

Antiderivative

An antiderivative of a function is essentially a function whose derivative gives back the original function. This is a central concept in calculus because finding antiderivatives allows for solving a range of real-world problems. An integral represents an antiderivative when there's no specified interval.
For the integral \( \int \frac{d x}{x^{2}} \), we find that the antiderivative is \(-\frac{1}{x} + C \).

• An antiderivative reverses differentiation.
• Used to compute areas under curves and solve differential equations.
• Commonly determined through various rules, including the Power Rule.

Each antiderivative is part of a more extensive family of similarly behaved functions, illustrated by the presence of the constant of integration \( C \), confirming that all potential antiderivatives are accounted for.