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Chapter 4: Problem 18

Determine these indefinite integrals. $$\int \frac{d x}{x^{2}}$$

Short Answer

Expert verified
\[ -\frac{1}{x} + C \]

Step by step solution

01

Set up the integral

Write the integral in its given form: \[ \int \frac{d x}{x^{2}} \]
02

Rewrite the integrand

Rewrite the integrand using the exponent rule for fractions. \[ \frac{1}{x^{2}} = x^{-2} \] Now the integral becomes: \[ \int x^{-2} \mathrm{d}x \]
03

Integrate using the power rule

Apply the power rule for integration, which states \int x^{n} \mathrm{d}x = \frac{x^{n+1}}{n+1} + C, where n \eq -1.\ Here, n = -2. Therefore, \[ \int x^{-2} \mathrm{d}x = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = \-x^{-1} + C \]
04

Simplify the result

Simplify the expression \-x^{-1} + C\. Since \x^{-1}\ can be rewritten as \frac{1}{x}\, the simplified form is: \[ -\frac{1}{x} + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrand
When solving indefinite integrals, the first thing you encounter is the integrand. This is the function you are integrating. In this exercise, the integrand is given as: \[ \frac{dx}{x^2} \]
To better handle this integral, we often need to transform the integrand into a friendlier form, which makes it easier to integrate.

By recognizing that \( \frac{1}{x^2} \) is the same as \( x^{-2} \), we can rewrite the integrand:

\[ \int x^{-2} dx \]
This transformation uses the exponent rule for fractions, which is a crucial algebraic technique in integration problems.
Power Rule for Integration
One of the key tools for solving indefinite integrals is the power rule for integration. It's a straightforward method to integrate functions of the form \( x^n \).

The power rule states that

\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
where \( n eq -1 \).

Applying the power rule to our example where \( n = -2 \), we integrate \( x^{-2} \):

\[ \int x^{-2} dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C \]
Simplifying further, we get:

\[ -x^{-1} + C \]
Simplification
After integrating using the power rule, the next step is simplification. Simplification helps us convert our answer into a more conventional or easily interpretable form.

When we have:

\[ -x^{-1} + C \]
we can simplify \( x^{-1} \) further since it equals \( \frac{1}{x} \).

Therefore, the simplified form of the integral becomes:

\[ -\frac{1}{x} + C \]
Simplification is crucial because it provides the final, polished form of our answer, ensuring it's as clear as possible.

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