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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: soccerboi on September 28, 2012, 04:24:42 pm

Title: When to include absolute value sign?
Post by: soccerboi on September 28, 2012, 04:24:42 pm: When antidiffing, how do you determine if the absolute sign is necessary?

e.g antiff (x+3)/(1+x²), why is it not needed here?

Thanks
Title: Re: When to include absolute value sign?
Post by: b^3 on September 28, 2012, 05:32:56 pm: Working through the problem.
$\begin{alignedat}{1}\int\frac{x+3}{1+x^{2}}dx & =\int\frac{x}{1+x^{2}}dx+3\int\frac{1}{1+x^{2}} \\ \mathrm{Let}\: u=1+x^{2} \\ \frac{du}{dx}=2x \\ \int\frac{x+3}{1+x^{2}}dx & =\frac{1}{2}\int\frac{1}{u}du+3\tan^{-1}(x)+C \\ & =\frac{1}{2}\ln(|x^{2}+1|)+3\tan^{-1}(x)+C \end{alignedat}$
Now as $x^{2}+1>0$ for all $x$ . So that means that the mod will have no affect on it, i.e. making a positive number positive is doing nothing.
So we have
$\int\frac{x+3}{1+x^{2}}dx=\frac{1}{2}\ln(x^{2}+1)+3\tan^{-1}(x)+C$
So basically to answer the actual question we can remove the modulus when the thing inside the modulus is always positive (or positive for a given domain if we restrict it).

So to check this, you can either
1) draw it out

2) Look at it logically, $x^{2}$ will always be positive as we square the number, then adding $1$ to it will make it remain positive, so $x^{2}+1>0$ for all $x$ .
Title: Re: When to include absolute value sign?
Post by: TommyLie on September 29, 2012, 11:56:20 am: Is it wrong to put a modulus on something which will be >0 anyway? Will you lose marks for it?
Title: Re: When to include absolute value sign?
Post by: BubbleWrapMan on September 29, 2012, 12:45:05 pm: Quote from: TommyLie on September 29, 2012, 11:56:20 am
Is it wrong to put a modulus on something which will be >0 anyway? Will you lose marks for it?
I wouldn't say so. You should be mindful of it though, I've seen a fair amount of questions that require the removal of the modulus signs to proceed, but in questions like soccerboi's example it shouldn't cost you a mark.

However, if you do remove the signs, it's a good idea to justify it (say something like "as x^2 + 1 > 0").