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There's a proof of that integral on this website:

I just don't understand why we need the absolute value.

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What's the natural log of $-1$? Notice that $\sin(x)=-1$ at infintely many points. – Todd Wilcox Jan 23 '13 at 23:48
The logarithm of negative numbers are imaginary numbers. – raindrop Jan 28 '13 at 17:17
up vote 4 down vote accepted

You cannot take the logarithm of a negative number.

Notice that $\dfrac{d}{dw} \log w = \dfrac1w$ (where $w$ is positive) and $\dfrac{d}{dw}\log(-w)$ (where $w$ is negative, so $-w$ is positive) is also $\dfrac1w$.

So $\dfrac{d}{dx}\log\sin x = \cot x$ for values of $x$ for which $\sin x$ is positive.

And $\dfrac{d}{dx}\log(-\sin x)=\cot x$ for value of $x$ for which $\sin x$ is negative, so that $-\sin x$ is positive.

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You want an answer that also works on an interval where $\sin x$ is negative.

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Compute the integral of $\cot(x)$ for $x\in[0,\pi]$ to be $\log(\sin(x))+C$. Then note that since $\cot(x)$ is an odd function, its integral will be an even function. Thus, the integral for $x\in[-\pi,\pi]$ will be $\log(\sin(|x|))+C$. By the periodicity, we get that the integral is $\log(|\sin(x)|)+C$.

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Isn't it $sin x$ and not $csc x$? Thanks! I love the use of odd and even functions. – raindrop Jan 24 '13 at 3:14
@Raindrop: you are right. Although $\log(\csc(x))$ is positive, $\log(\sin(x))$ is increasing. – robjohn Jan 24 '13 at 3:24
I don't get it. The formal answers (the answer key of my textbook and the website link in my question) say the answer is ln sin x. The answers on this website say the answer is log sin x. How could ln and log be used interchangeabley? Is it somehow assumed that I know you are referring to $log_{e}$ and not just $log_{10}$? I understand that the natural log refers to ln a.k.a. $log_{e}$. – raindrop Jan 24 '13 at 4:35
@Raindrop: if I'm talking to a mathematician, I assume all $\log$ and $\ln$ are natural logs, unless otherwise specified. When I was in grad school, engineers would use $\log$ for common and $\ln$ for natural. I apologize for any confusion. In my posts, I use $\log$ and mean natural log. – robjohn Jan 24 '13 at 5:17

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