One more antiderivative question.

[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 separately, 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.

 

[arildno]

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

 

[Checkfate]

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?

 

[arildno]

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

 

[radou]

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.

 

[Checkfate]

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

Thanks!

 

[arildno]

That is indeed correct.

 

[radou]

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).

 

[Checkfate]

True, thanks. =)