What Is the Correct Antiderivative of 1/(x^2)?

In summary, an antiderivative is the reverse operation of a derivative and can be used to solve problems involving rates of change. The process for finding an antiderivative involves using integration rules and techniques, and not all functions have an antiderivative. The antiderivative is related to the definite integral through the Fundamental Theorem of Calculus.
  • #1
bobsmith76
336
0

Homework Statement



find the antiderivative of 1/(x^2)


The Attempt at a Solution



I'm pretty sure you just find the antiderivatives of the numerator and the denominator.

the antiderivative of 1 is x.
the antiderivative of x^2 is (x^3)/3
mutliply the numerator by the inverse of the deominator and you get 3/(x^2)

The book says the answer is -1/x
 
Physics news on Phys.org
  • #2
bobsmith76 said:

Homework Statement



find the antiderivative of 1/(x^2)

The Attempt at a Solution



I'm pretty sure you just find the antiderivatives of the numerator and the denominator.

the antiderivative of 1 is x.
the antiderivative of x^2 is (x^3)/3
mutliply the numerator by the inverse of the deominator and you get 3/(x^2)

The book says the answer is -1/x

Have you ever seen the result ∫ x^n dx = x^(n+1)/(n+1) + C? It does not hold for n = -1 because that would involve division by zero on the right-hand-side, but otherwise n is unrestricted.

Please, please, please *get rid of the idea forever* that you can do the integral by integrating the numerator and denominator separately: that does not work! For example, integrate x, to get x^2/2. Now write x as x^2 / x and try your method of integrating the numerator and denominator separately---you will get the wrong answer.

RGV
 
Last edited:
  • #3
... (posted an idea that was false, subsequently deleted it)
 
  • #4
Then how do I find the antiderivative of a fraction?

Here's another problem

1. 2/3 sec^2 x/3

2. (2/3 tan x/3)/(x/3)

3. (2/3 tan x/3) * (3/x)

4. 2/3 * 3/x (tan x/3)

5. 2/x tan x/3

the book says the answer is

2 tan (x/3)
 
  • #5
Ok, I saw from another website, that to find the antiderivative of a fraction you have to convert the fraction into a number with an exponent.

so -1/x = -x^-1

apply (x^n+1)/(n+1)

that comes to

-(x^0)/0
 
  • #6
bobsmith76 said:
Then how do I find the antiderivative of a fraction?

Here's another problem

1. 2/3 sec^2 x/3

2. (2/3 tan x/3)/(x/3)

3. (2/3 tan x/3) * (3/x)

4. 2/3 * 3/x (tan x/3)

5. 2/x tan x/3

the book says the answer is

2 tan (x/3)
Assuming that you mean to find the anti-derivative of (2/3) sec2 (x/3) :

Rather than trying to follow those steps, which I can't make sense of ... Let's see if the book's answer is correct.

If 2 tan(x/3) is an anti-derivative of (2/3) sec2 (x/3), then the derivative of 2 tan(x/3), with respect to x, should be (2/3) sec2 (x/3) .

[itex]\displaystyle \frac{d}{dx}\tan(x)=\sec^2(x)[/itex]

So that: [itex]\displaystyle \frac{d}{dx}\left(2\,\tan\left(\frac{x}{3}\right) \right)=2\,\sec^2\left(\frac{x}{3}\right)\cdot \frac{1}{3}=\frac{2}{3}\, \sec^2\left(\frac{x}{3}\right)\,.[/itex]

The book's answer is correct.
 
  • #7
Knowing the answer than finding the derivative can be done, but I still don't see how to find the answer starting from the antiderivative.

I got the answer to 1/x^-2 now.
 
  • #8
bobsmith76 said:
Ok, I saw from another website, that to find the antiderivative of a fraction you have to convert the fraction into a number with an exponent.

so -1/x = -x^-1

apply (x^n+1)/(n+1)

that comes to

-(x^0)/0
The anti derivative of xn=xn+1/(n=1), except if n = -1.

Why that exception?
It's because x0=1, and the derivative of 1 is 0, not x-1.​

The derivative of x-1 = ln(|x|)
 
  • #9
bobsmith76 said:
Knowing the answer then finding the derivative can be done, but I still don't see how to find the answer starting from the antiderivative.

I got the answer to 1/x^-2 now.
The anti-derivative of 1/x-2 is easy, because 1/x-2 = x2.

Earlier, you wanted the anti-derivative of 1/x2, which you found to be -1/x . That's easy to get if you write 1/x2 as x-2 .

Details ... details.
 
  • #10
Did you fail to read my first post, which specifically told you this result fails for n = -1?

RGV
 

Related to What Is the Correct Antiderivative of 1/(x^2)?

What is an antiderivative?

An antiderivative, also known as the indefinite integral, is the reverse operation of a derivative. It is a function that, when differentiated, produces a given function.

Why is finding the antiderivative important?

Finding the antiderivative is important because it allows us to solve problems involving rates of change, such as calculating the area under a curve or finding the velocity of an object at a specific time.

What is the process for finding the antiderivative?

The process for finding the antiderivative involves using integration rules and techniques, such as substitution and integration by parts, to manipulate the given function into a form that can be integrated. It may also involve using known antiderivatives for common functions.

Can all functions have an antiderivative?

No, not all functions have an antiderivative. Functions that are continuous and have a finite derivative at every point in their domain have an antiderivative. However, some functions, such as the Dirichlet function, do not have an antiderivative.

How is the antiderivative related to the definite integral?

The antiderivative and the definite integral are related through the Fundamental Theorem of Calculus, which states that the definite integral of a function can be calculated by finding its antiderivative and evaluating it at the limits of integration. In other words, the definite integral is the difference between the antiderivative evaluated at the upper and lower limits of integration.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Replies
20
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Replies
8
Views
494
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Back
Top