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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: monkeywantsabanana on October 09, 2011, 07:57:19 pm

Title: Change of Variable
Post by: monkeywantsabanana on October 09, 2011, 07:57:19 pm: Hey guys,

When anti deriving, when do you know when to use the Change of Variable (Substitution Method)? Is there a way to recognise these questions and where to apply the method?

Cheers.
Title: Re: Change of Variable
Post by: jane1234 on October 09, 2011, 08:06:24 pm: Quote from: monkeywantsabanana on October 09, 2011, 07:57:19 pm
Hey guys,

When anti deriving, when do you know when to use the Change of Variable (Substitution Method)? Is there a way to recognise these questions and where to apply the method?

Cheers.

Yes, I used to constantly struggle with these. I used to always spend about half the exam time tring to use substitution only to realise it could have been done another way in about 5 mins.
What I do is use it as a last resort.
If you can't use one of the following techniques:

- Recognition of the derivative in the integrand
- Splitting into partial fractions
- Using double angle formulas
- Recognition that it is a derivative of inverse sin(x), cos(x) or tan(x).

THEN you try substitution. It doesn't take long to mentally go through these 4 techniques, and if none of them will work try substitution.

Also, note that substitution is almost always used when a LINEAR function is under a square root sign or raised to a power of something. You can use substitution for recognition questions, but I find it quicker skip that step, then derive what you have gotten to check that it is right.
Title: Re: Change of Variable
Post by: tony3272 on October 09, 2011, 08:07:45 pm: The first thing i look for is whether or not the question fits any of the standard equations we have. If it doesn't then i look for partial fractions. If it isn't either of them it's pretty much a substitution.

As you do more exams you'll get used to the type of questions which you need to use it or not.

*Edit*: yeah as Jane said, use it as a last resort. It's time consuming and you can confuse yourself a lot more if you try it in the wrong situation.
Title: Re: Change of Variable
Post by: b^3 on October 09, 2011, 08:09:12 pm: They way I normally spot it, and this is only for polynomial functions is if the bottom terms has a highest power that is one higher than the highest power of the terms on the top. If that occurs then odds are that the derivative of the bottom will cancel out the terms on the top (when you use substituion) and leave a constant over.
Title: Re: Change of Variable
Post by: tony3272 on October 09, 2011, 08:12:56 pm: Quote from: b^3 on October 09, 2011, 08:09:12 pm
They way I normally spot it, and this is only for polynomial functions is if the bottom terms has a highest power that is one higher than the highest power of the terms on the top. If that occurs then odds are that the derivative of the bottom will cancel out the terms on the top (when you use substituion) and leave a constant over.

b^3 are you just talking about this rule? $\int \frac{f'(x)}{f(x)}dx=log_{e}\left |f(x) \right |+c$
Title: Re: Change of Variable
Post by: b^3 on October 09, 2011, 08:26:37 pm: Nah I mean like say for example find $\int \frac{-4x-2}{\sqrt{x^{2}+x}}dx$
the let $u=x^{2}+x, \frac{du}{dx}=2x+1$
$dx=\frac{1}{2x+1}du$
$\int \frac{-4x-2}{\sqrt{x^{2}+x}}dx=\int \frac{-2(2x+1)}{u^{\frac{1}{2}}}\frac{1}{2x+1}du$
$=\int \frac{-2}{u^{\frac{1}{2}}}du$
$=-2*-2u^{\frac{1}{2}}$
$=4u^{\frac{1}{2}}$
$=4\sqrt{x^{2}+x}$
Title: Re: Change of Variable
Post by: tony3272 on October 09, 2011, 08:28:02 pm: ah okay that makes more sense.
Title: Re: Change of Variable
Post by: monkeywantsabanana on October 10, 2011, 06:13:12 pm: thanks guys !
Title: Re: Change of Variable
Post by: lorelai on October 12, 2011, 08:44:28 pm: I've noticed in practice exams and stuff that the examiners are heading towards finding the expression for substitution and choosing the correct answer in multiple choice, rather than finding the full antiderivative in the extended answer/short answer questions. This is handy because from the options listed you can immediately recognise that they've used this rule because they have everything in terms of δu instead of δx. Then all you have to do is to recognise the correct substitution and find the correct limits.

Probably something you already knew, but meh.
Title: Re: Change of Variable
Post by: costa94 on October 12, 2011, 08:57:06 pm: outrageously obvious if you actually do spec math
this just shows me that you're a chump

...but seriously,
DO MATH!!!