One more antiderivative question.

In summary, the conversation is about finding the antiderivative of \frac{1}{2x}. The individual is trying to separate the terms in order to solve each antiderivative separately, but is unsure how to do so. They mention converting \frac{1}{2x} to 2x^{-1}, but realize that this is not possible since there is no + or - between the terms. Another individual suggests using \int \frac{1}{x}dx = \ln \left|x \right|+C as a typical "table integral" and the solution is \frac{1}{2}*ln(|x|) + C. The original individual acknowledges this and adds that the constant of integration, C
  • #1
Checkfate
149
0
Hi there, I am trying to find the antiderivative of [tex]\frac{1}{2x}[/tex] but can't seem to do it. I am trying to separate the terms so that I can do each antiderivative seperately, but I don't see a way to do that.

My most natural first attempt is to convert it to [tex]2x^{-1}[/tex] but of course since they are not separated by a + or minus (the 2 and the x^-1) I can't use antiderivative laws on it. If I did, I would end up with [tex] 2x*ln(\abs{x})[/tex] which is wrong. Any help is appreciated, thanks.

PS- I have not learned to integrate by parts or anything like that yet.
 
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  • #2
[tex]\frac{1}{2x}=\frac{1}{2}*\frac{1}{x}=a*\frac{1}{x}=\frac{a}{x}, a=\frac{1}{2}[/tex]
Does that help?
 
  • #3
Woops sorry, some bad algebra there. I meant [tex]\frac{1}{2}*x^{-1}[/tex].

But no, I don't think it does, because I would get [tex]\frac{x}{2}*ln(|x|)[/tex] wouldn't I? (Assuming you mean to just use the antiderivative formulas immediately) :(. What am I doing wrong?
 
  • #4
What is the derivative of g(x)=a*f(x), where "a" is a constant?
 
  • #5
Checkfate said:
PS- I have not learned to integrate by parts or anything like that yet.

You don't need to know how to integrate by parts to solve this, since [tex]\int \frac{1}{x}dx = \ln \left|x \right|+C[/tex] is a typical 'table integral'. You'll find the integral of [tex]f(x) = \frac{1}{2x}[/tex] easily now by reading arildno's comments.
 
  • #6
Ahh kk I get it, [tex]=\frac{1}{2}*ln(|x|)[/tex] :)

Thanks!
 
  • #7
That is indeed correct.:smile:
 
  • #8
Checkfate said:
Ahh kk I get it, [tex]=\frac{1}{2}*ln(|x|)[/tex] :)

Thanks!

Correct, but don't forget to add the constant of integration, C (or whatever you like to name it). :smile:
 
  • #9
True, thanks. =)
 

Related to One more antiderivative question.

1. What is an antiderivative?

An antiderivative, also known as the indefinite integral, is the reverse process of finding the original function from its derivative. It is denoted by adding a "+ C" at the end of the integral symbol.

2. Why is it important to find antiderivatives?

Finding antiderivatives allows us to solve many real-world problems, such as calculating the area under a curve or determining the velocity of an object at a specific time. It also helps us understand the behavior of functions and their relationships with each other.

3. How do you find an antiderivative?

To find an antiderivative, you need to use integration techniques such as u-substitution, integration by parts, or trigonometric substitution. It also helps to have a good understanding of basic integration rules and properties.

4. Is there more than one antiderivative for a given function?

Yes, there can be an infinite number of antiderivatives for a given function. This is because any constant value can be added to the antiderivative, and it will still satisfy the condition of being the reverse process of differentiation.

5. Can every function have an antiderivative?

No, not every function has an antiderivative. This is because some functions are not continuous or do not have a continuous derivative, which is a necessary condition for a function to have an antiderivative.

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