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# C4 Integration by substitution, definite watch

1. When we substitute x = tan (theta) to find definite integration, why we assume theta to be an acute angle?
2. Would need context but the only reason I can think of off the top of my head is that tan is positive when acute and negative 90<theta<180
3. (Original post by mystreet091234)
When we substitute x = tan (theta) to find definite integration, why we assume theta to be an acute angle?
We don't? We simple restrict to be piecewise injective over our integration domain. This means that we need (or any other interval of length , with the obvious exception of in which case we get an improper integral. You might want to give an example or such so that we can clarify.
4. (Original post by Zacken)
We don't? We simple restrict to be piecewise injective over our integration domain. This means that we need (or any other interval of length , with the obvious exception of in which case we get an improper integral. You might want to give an example or such so that we can clarify.
How do we know if the upper limit is not pi-pi/6 for obtuse angle?
Q:
A:Attachment 513987513989
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5. (Original post by mystreet091234)
How do we know if the upper limit is not pi-pi/6 for obtuse angle?
Because, like I said - we need to be an injective mapping, i.e: can only have one value and we pick the principal value .
6. (Original post by Zacken)
Because, like I said - we need to be an injective mapping, i.e: can only have one value and we pick the principal value .
I see thanks!
7. (Original post by Zacken)
We don't? We simple restrict to be piecewise injective over our integration domain. This means that we need (or any other interval of length , with the obvious exception of in which case we get an improper integral. You might want to give an example or such so that we can clarify.
Sorry I'm still confused. Why does it have to be injective?

For example, why isn't there a plus/minus sign near the u in step (iii) ?

Also for finding Cartesian equation of a curve from its parametric equations x=sin t and y=sin 2t, are the two equations also injective mapping?

But for another question, it seems the marking scheme isnt treating the equation as injective...when should an equation be treated as injective?
Attachment 514541514543

Sorry for my dumbness
Zacken aymanzayedmannan
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8. (Original post by mystreet091234)
Sorry for my dumbness
You're not being dumb - it's actually a really good thing that you're attempting to understand why we're using a one-to-one function rather than just learning rote methods.

(Original post by mystreet091234)
But for another question, it seems the marking scheme isnt treating the equation as injective...when should an equation be treated as injective?
However, for this question I don't really see anything about restricting domains as such, this seems like straightforward differentiation.
9. (Original post by aymanzayedmannan)
You're not being dumb - it's actually a really good thing that you're attempting to understand why we're using a one-to-one function rather than just learning rote methods.

However, for this question I don't really see anything about restricting domains as such, this seems like straightforward differentiation.
I'm waiting for Zacken's reply too
You two are the besttt

So for that question when we don't have to restrict domain, one-to-one function isn't needed?

When do we need to restrict domain? for integration and parametric equation?
10. (Original post by mystreet091234)
Sorry I'm still confused. Why does it have to be injective?
It doesn't really need to be injective, it just needs to be piece wise injective. So for example, I can take piecewise on and where it is injective on these two different intervals separately. This is a bit hard to wrap your head around, I know - so I'm going to say that at this level, you can treat mostly everything as "just taking the positive root" - so if you have , don't worry about it - at A-Level you will only just need to take the positive root and write down . I wish I had a better explanation but I don't have quite enough time to type one out right now. Don't worry about injectivity, it seems that I have just confused you by using the word.
11. (Original post by Zacken)
It doesn't really need to be injective, it just needs to be piece wise injective. So for example, I can take piecewise on and where it is injective on these two different intervals separately. This is a bit hard to wrap your head around, I know - so I'm going to say that at this level, you can treat mostly everything as "just taking the positive root" - so if you have , don't worry about it - at A-Level you will only just need to take the positive root and write down . I wish I had a better explanation but I don't have quite enough time to type one out right now. Don't worry about injectivity, it seems that I have just confused you by using the word.
Okay so for sub-fuctions of the piecewise function just take positive roots so this also explains my parametric equation problem!!!

THANKS SO MUCH
12. (Original post by mystreet091234)
I'm waiting for Zacken's reply too
You two are the besttt

So for that question when we don't have to restrict domain, one-to-one function isn't needed?

When do we need to restrict domain? for integration and parametric equation?
One way to avoid thinking about it is by taking a sub of . We know that squaring both sides only yields the positive value of the function. Honestly, don't worry about restriction of domains because you'll only ever need the positive root in A-level maths and further maths.
13. (Original post by aymanzayedmannan)
One way to avoid thinking about it is by taking a sub of . We know that squaring both sides only yields the positive value of the function. Honestly, don't worry about restriction of domains because you'll only ever need the positive root in A-level maths and further maths.
Got it!!
Thank you guys
14. (Original post by mystreet091234)
Okay so for sub-fuctions of the piecewise function just take positive roots so this also explains my parametric equation problem!!!

THANKS SO MUCH
Here's a post I've made about injectivity and substitution in the past: http://www.thestudentroom.co.uk/show...9#post62689379 - you might find it vaguely illuminating.

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