# One more antiderivative question.

• Checkfate
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
Checkfate
Hi there, I am trying to find the antiderivative of $$\frac{1}{2x}$$ 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 $$2x^{-1}$$ 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 $$2x*ln(\abs{x})$$ which is wrong. Any help is appreciated, thanks.

PS- I have not learned to integrate by parts or anything like that yet.

$$\frac{1}{2x}=\frac{1}{2}*\frac{1}{x}=a*\frac{1}{x}=\frac{a}{x}, a=\frac{1}{2}$$
Does that help?

Woops sorry, some bad algebra there. I meant $$\frac{1}{2}*x^{-1}$$.

But no, I don't think it does, because I would get $$\frac{x}{2}*ln(|x|)$$ wouldn't I? (Assuming you mean to just use the antiderivative formulas immediately) :(. What am I doing wrong?

What is the derivative of g(x)=a*f(x), where "a" is a constant?

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 $$\int \frac{1}{x}dx = \ln \left|x \right|+C$$ is a typical 'table integral'. You'll find the integral of $$f(x) = \frac{1}{2x}$$ easily now by reading arildno's comments.

Ahh kk I get it, $$=\frac{1}{2}*ln(|x|)$$ :)

Thanks!

That is indeed correct.

Checkfate said:
Ahh kk I get it, $$=\frac{1}{2}*ln(|x|)$$ :)

Thanks!

Correct, but don't forget to add the constant of integration, C (or whatever you like to name it).

True, thanks. =)

## 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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