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Chapter 1: Problem 7

In the following exercises, compute each indefinite integral. $$ \int \frac{1}{x^{2}} d x $$

Short Answer

Expert verified
\(-\frac{1}{x} + C\) is the indefinite integral.

Step by step solution

01

Identify the Integral Form

First, recognize that the integral \( \int \frac{1}{x^2} \, dx \) can be rewritten using a negative exponent: \( \int x^{-2} \, dx \). This will allow us to use the power rule for integration.
02

Apply the Power Rule for Integration

The power rule for integration states that for \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) when \( n eq -1 \). In this case, \( n = -2 \), so applying the rule gives \( \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C \).
03

Simplify the Expression

The expression \( \frac{x^{-1}}{-1} + C \) simplifies to \( -x^{-1} + C \). Since \( x^{-1} = \frac{1}{x} \), we can rewrite the integral as \( -\frac{1}{x} + C \).
04

Combine Results

The indefinite integral of the original function \( \int \frac{1}{x^2} \, dx \) is now written as \( -\frac{1}{x} + C \), where \( C \) is the constant of integration.

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Key Concepts

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

Power Rule for Integration
The power rule for integration is a simple and powerful tool used to find antiderivatives. When you see an integral of the form \( \int x^n \, dx \), you can apply this rule where \( n eq -1 \). The rule states:
  • If you have \( \int x^n \, dx \), it simplifies to \( \frac{x^{n+1}}{n+1} + C \).
  • Here, \( n+1 \) becomes the new exponent.
  • You also divide by \( n+1 \) to balance the equation.
This rule is crucial because it transforms the integration process into a straightforward calculation by adjusting the exponent and introducing a division. The rule doesn鈥檛 work when \( n = -1 \) because it would lead to division by zero, so a different approach is needed in those cases.
Negative Exponent
Sometimes, functions look complicated, like when you encounter \( \frac{1}{x^2} \). Luckily, negative exponents simplify this process:
  • Rewrite the fraction using a negative exponent: \( x^{-2} \).
  • This fits perfectly with the power rule for integration.
  • Now, integrate as if it's a normal power of \( x \).

Negative exponents are especially helpful as they often simplify complex fractions, making it easier to integrate using typical rules like the power rule. A habit of rewriting using negative exponents will enhance your ability to solve integrals quickly and accurately.
Constant of Integration
When performing indefinite integration, it鈥檚 crucial to remember the constant of integration, denoted by \( C \). This constant is important because:
  • An integral represents a family of functions, not just one single function.
  • Without \( C \), you would miss many possible solutions that differ by a constant.
  • In practical terms, the constant encompasses all possible initial conditions or vertical shifts in the graph of an antiderivative.
Always add \( C \) to your solution to acknowledge the full family of antiderivatives, ensuring you account for all potential solutions in problems involving curves and slopes.

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