One more antiderivative question.

Checkfate

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.

arildno

[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?

Checkfate

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?

arildno

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

radou

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.

Checkfate

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

Thanks!!!

arildno

That is indeed correct.

radou

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

Checkfate

True, thanks. =)