Indefinite Integral of (1/x^2)

In summary, the book says to use the reverse power rule to find (-1/x), but the problem doesn't make sense because n=-2. Somebody please help me understand this!
  • #1
Liger20
65
0

Homework Statement



Hello, first of all I would like to apologize for the fact that this question is extremely trivial compared to the other questions being asked. I have a improper integral problem, and the entire problem itself is not relevant, because I understand everything in it except for one thing. One step in the problem requires finding the indefinite integral of (1/x^2). The example in the book tells me that the answer is (-1/x), and it says that the answer is obtained by using the reverse power rule, but I just can’t see how they got that answer. I have a feeling that it is something very simple, and that I’ve forgotten some subtle detail.



Homework Equations








The Attempt at a Solution




Here’s how I tried to solve it:

(1/x^2) has an overall power of one, right? I increased the power of the whole thing by one, which is (1/x^2)^2, and I divided the whole thing by two, which is the same thing as multiplying by ½. So…

(½)(1/x^2)^2

You can already see that this is not going to give an answer of (-1/x). Could someone please help me with this?
 
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  • #2
[tex]\int x^n\,dx = {1\over n+1}x^{n+1} + C[/tex]
In your case, [itex]n=-2[/itex].
 
  • #3
Hmmm...okay, I'll buy that. But why is n negative 2? If I'm raising the whole thing to the power of 2, why isn't n just positive 2? Thank you!
 
  • #4
[tex]
\frac{1}{x^n} = x^{-n}
[/tex]

A basic property of exponents.
 
  • #5
Raising the entire expression to the power of 2 would give you [itex](1/x^2)^2=1/x^4=x^{-4}[/itex].
 
  • #6
Liger20 said:
Hmmm...okay, I'll buy that. But why is n negative 2? If I'm raising the whole thing to the power of 2, why isn't n just positive 2? Thank you!
To expand on what kbaumen wrote, and relative to your problem,
[tex]\frac{1}{x^2} = x^{-2}[/tex]
 

What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the antiderivative, or the inverse operation, of differentiation. It is a function that, when differentiated, gives the original function.

What is the indefinite integral of (1/x^2)?

The indefinite integral of (1/x^2) is -1/x + C, where C is a constant of integration.

How do you solve an indefinite integral?

To solve an indefinite integral, you can use the power rule, which states that the integral of x^n is (x^(n+1))/(n+1) + C. For fractions, you can use the rule for integration by substitution, where you substitute u for the denominator and du for the numerator.

What is the significance of the constant of integration?

The constant of integration, denoted as C, represents an infinite number of possible solutions to the indefinite integral. This is because when you differentiate a constant, it becomes 0. Therefore, the constant of integration allows for all possible solutions to the indefinite integral.

How is the indefinite integral of (1/x^2) used?

The indefinite integral of (1/x^2) is used in various applications in physics, engineering, and economics. It can represent the velocity of an object with constant acceleration, the electric field of a point charge, and the marginal cost in economics, among other things.

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