Maximus 190
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For this question, I am not too sure how they worked out the different form of the integral where they say 'Using the standard method of changing variable in an integral (basically the chain rule), the integral
becomes:'. I can see that if you cancel out the dt's you get f(t)dx, however I'm pretty sure that isn't proper maths and doesn't always work. I've always struggled with understanding differentiation/integration so would appreciate some help here
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Qxi.xli
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Don't know why I clicked on this thread thinking I could help😳..

Anways at least ill bump this up for you .
Last edited by Qxi.xli; 1 month ago
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JGLM
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So if you use their substitution you have to find dt/dx and find dx in terms of t, then substitute back in. Should fall out. I’ll give it a try to make sure
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DFranklin
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(Original post by Maximus 190)
Question:
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Start of solution:
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For this question, I am not too sure how they worked out the different form of the integral where they say 'Using the standard method of changing variable in an integral (basically the chain rule), the integral
becomes:'. I can see that if you cancel out the dt's you get f(t)dx, however I'm pretty sure that isn't proper maths and doesn't always work. I've always struggled with understanding differentiation/integration so would appreciate some help here
This is, literally, the standard way of doing an integral by substitution. They've not done anything different (other than explain it in a somewhat confusing manner).
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Maximus 190
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(Original post by DFranklin)
This is, literally, the standard way of doing an integral by substitution. They've not done anything different (other than explain it in a somewhat confusing manner).
Yeah okay I follow it now. Thanks
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JGLM
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Lol, it’s the dx/dt part, getting x in terms of t is an absolute nightmare... Is this STEP 3?
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DFranklin
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(Original post by Maximus 190)
Yeah okay I follow it now. Thanks
The "tricky bit" is finding dt/dx in terms of t. I don't recall how Siklos does it in the guide, but a useful observation here that comes up quite often in STEP is:

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(\sqrt{A+1} + \sqrt{A})(\sqrt{A+1} - \sqrt{A}) = 1
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DFranklin
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(Original post by JGLM)
Lol, it’s the dx/dt part, getting x in terms of t is an absolute nightmare... Is this STEP 3?
Pretty certain it's STEP I or STEP II. You see similar manipulations when mucking around with hyperbolics, so it would be a bit more familiar if you were doing FM.
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JGLM
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(Original post by DFranklin)
Pretty certain it's STEP I or STEP II. You see similar manipulations when mucking around with hyperbolics, so it would be a bit more familiar if you were doing FM.
I’ve managed to get it out now, it was an easy spot that I was overlooking, nothing to do with hyperbolics.
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DFranklin
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(Original post by JGLM)
I’ve managed to get it out now, it was an easy spot that I was overlooking, nothing to do with hyperbolics.
It's not directly to do with hyperbolics, but if you look at the formulae for arsinh and arcosh you can see the similarities. What I was getting at is if you've done a fair amount of work with hyperbolics the manipulations you need here to write everything in terms of t should be much more familiar.

Edit: (and somewhat as a consequence, questions involving manipulating sqrt(x^2+1) +/- x tend to be very much STEP I/II rather than III).
Last edited by DFranklin; 1 month ago
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JGLM
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(Original post by DFranklin)
It's not directly to do with hyperbolics, but if you look at the formulae for arsinh and arcosh you can see the similarities. What I was getting at is if you've done a fair amount of work with hyperbolics the manipulations you need here to write everything in terms of t should be much more familiar.

Edit: (and somewhat as a consequence, questions involving manipulating sqrt(x^2+1) +/- x tend to be very much STEP I/II rather than III).
Yeah I see what you’re saying. Having done it it feels much more like a hard STEP 1 than a STEP 2. You just have to know your way around the algebra lol
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