Stupid old me.
Here is a past paper question that I can't seem to get out. I tried integration by parts but the mark scheme suggests using substitution. Now I know substitution but I can't just figure out how to use it here.
Sometimes I feel so dumb.
Maybe keep in mind that 1+t^2=sec^2
I know that identity.
But how do I use that?
Perform the substitution, and get sec^4 in the denominator (which I assume you've got to), then the way I did is probably not the easiest (if it is correct!), but try dividing through sec^2 and then by cos^2 and see what you get, I'm doing this without paper and pen so may be making algebraic slips.
Thanks for the attachement. However, I cannot see the workings on the page - they are giving me some quicktime error.
May I ask what it is you used? Quicktime? What do I need to download to make this work?
This is another attachment - I think this has the solution. What I am concerned with is the logic when changing the variable. I read 2 or three pages on the matter and I made this. Is my logic okay? Is this how we are supposed to do this stupid change of variable thing?
Hit me back guys (especially you martinkings - I am anxious to know what format you used. I use microsoft's equation editor).
It's a long story about going abroad and forgetting my laptop - but I'm using a Mac at the moment, and I've just used word on it, I simply used a latex program and dragged the pics into the doc.
It looks fine from my side - but if anyone out there uses a word and a mac - tell me what to do.
Why would we do that subbing back into the expression?
I never thought about it like that.
correct me if I am wrong but we are using this thing:
F(u) * du/dx = F(u) du
I'm not entirely sure what you mean with that F(u)....
I assume you're on about changing dx to du/d(theta)?
This is an identity my maths text (bostock chandler) has.
Is this what we have to "keep in mind" as we change a variable (I am still trying to understand the concept behind this change of variable thing.)
Forget these ‘identities,’ you just need to remember:
When you use a substitution, you’re saying let x (or whatever) = some other variable; you therefore need to change what you’re integrating with respect to also, in order for it to ‘balance’ if you like.
The original question was ‘dx’ – which you know means integrate with respect to x. If you now say that x = tan u, you need to integrate with respect to the new variable, u – hence you need dx to become du.
So in this case, it was x = tanθ, thus we need dθ instead of dx.
x = tanθ
dx/dθ = sec2θ
This is what you need to re-arrange like any other equation to make ‘dθ’ the subject in order to replace the dx in the original integral.
I’m not sure if I’ve made that very clear or been as mathematically accurate as need be, but I hope it helps...
Indeed. I pasted from word & 4got the '^'. Sadly I cannot get proper maths lingo 2 work on here. hmm...
Got you there Paul Mitchell.
That explanation was concise and clear. As a matter of fact, I believe it was this identity that was causing my confusion in the first place; I used your method on some of the other advanced integration in my text and everything seems to work out fine.
All that maths made me hungry.
What do the English much on when they figured out a new mathematical technique?
Where are my manners?
Thanks guys for all your help.
With you guys I am sure I can pull of an A.