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OCR Core 4 Intergration help

Hi guys, I am stuck on this question and needed some help!

∫1/x2+4 dx\int 1/x^2 +4 \ dx

The limits are x=infinity and x=0

Using the subsitution x=2tanu
therefore x^2+4 = 4tan^2u +4
= 4(tan^2u+1)
= 4 sec^u
dx/du= 2sec^2u

Now the limits is where I am stuck on...

If we treat infinity as s, then u=tan^-1 (s/2)
and also u=0 for the other x value.

Now we get the integral of 1/2
as we can cancel the differntial and the 4sec^2 u down...

so this becomes [u/2] with limits tan^-1 (s/2) and 0

I got the question wrong as the book says its 1/4pi wtf?

Help please:smile:

P.S THE initial integral is 1 divided by (x^2+4) to avoid confusion
(edited 13 years ago)
Reply 1
Avatar for J DOT A
J DOT A
OP
bump!
Original post by J DOT A


tan^-1 (s/2)


Well what does that come out as?
Reply 3
Avatar for J DOT A
J DOT A
OP
Original post by ghostwalker
Well what does that come out as?


tan^-1 (s/2)?
surely you have to leave it in that form? Or am I missing something stupidly obvious? Would you treat S as 1?

I am confused:frown:
Original post by J DOT A
tan^-1 (s/2)?
surely you have to leave it in that form? Or am I missing something stupidly obvious? Would you treat S as 1?

I am confused:frown:


You said in your initial post you were using "s" for infinity.

You have two choices regarding the integral: Either convert the limits to be in terms of u, and substitute into your result, OR convert the result back into terms of x and use your original limits, but it makes little difference to the working.
(edited 13 years ago)
Reply 5
Avatar for J DOT A
J DOT A
OP
Original post by ghostwalker
You said in your initial post you were using "s" for infinity.

You have two choices regarding the integral: Either convert the limits to be in terms of u, and substitute into your result, OR convert the result back into terms of x and use your original limits, but it makes little difference to the working.



Thats what I did, convert the limits to be in terms of u...? But I still have no clue what to do.
Original post by J DOT A
Thats what I did, convert the limits to be in terms of u...? But I still have no clue what to do.


Well if you've converted them you should have π2\frac{\pi}{2} and 0, and now just sub into your function u/2, and you get π4\frac{\pi}{4} as required.
sorry edited out stupid comment
(edited 13 years ago)
Here is how I did it using mathematica, please query anything you don't understand, i'll add a step explanation later if you like

EDIT: using a t=tan[x/2] substitution
Untitled-1.pdf108.8KB
Reply 9
Avatar for J DOT A
J DOT A
OP
Original post by ghostwalker
Well if you've converted them you should have π2\frac{\pi}{2} and 0, and now just sub into your function u/2, and you get π4\frac{\pi}{4} as required.


Sorry to keep annoying you:tongue: but can I ask how you got pi/2 by subsitituion?
Could you show me step by step please?
Original post by J DOT A
Sorry to keep annoying you:tongue: but can I ask how you got pi/2 by subsitituion?
Could you show me step by step please?




tan⁡−1∞=lim⁡x→∞tan⁡−1x=π2\displaystyle \tan^{-1}\infty = \lim_{x\to\infty}\tan^{-1}x= \frac{\pi}{2}


Where tan⁡−1\tan^{-1} is defined with domain R\mathbb{R} and codomain [−π2,π2][-\frac{\pi}{2},\frac{\pi}{2}], which is effectively what you set up when you defined the substitution.



Note: Anyone well up on uni. maths feel free to correct, as I'm somewhat rusty on limits. I think that's good enough for A-level though.
(edited 13 years ago)
Reply 11
Avatar for ElMoro
ElMoro
18
Original post by ghostwalker
tan⁡−1∞=lim⁡x→∞tan⁡−1x=π2\displaystyle \tan^{-1}\infty = \lim_{x\to\infty}\tan^{-1}x= \frac{\pi}{2}


Where tan⁡−1\tan^{-1} is defined with domain R\mathbb{R} and codomain [−π2,π2][-\frac{\pi}{2},\frac{\pi}{2}], which is effectively what you set up when you defined the substitution.



Note: Anyone well up on uni. maths feel free to correct, as I'm somewhat rusty on limits. I think that's good enough for A-level though.


When I did core 4, I didn't do any limits like this. :erm:
Will I need it for STEP?
Original post by ElMoro
When I did core 4, I didn't do any limits like this. :erm:
Will I need it for STEP?

It's nothing too special, you just need knowledge of the arctan function for this limit (Since you're on OCR, that was probably seen in C3) and the idea of improper integrals (C2) to finish the job.

You don't need too much more than what you've covered in the A-Level for STEP. Off the top of my head, additional things to know are Maclaurin series, proof by induction, sketching the graphs of rational functions and probably a few others that you'll easily spot if you look at their syllabus for STEP I.
Reply 13
Avatar for ElMoro
ElMoro
18
Original post by Farhan.Hanif93
It's nothing too special, you just need knowledge of the arctan function for this limit (Since you're on OCR, that was probably seen in C3) and the idea of improper integrals (C2) to finish the job.

You don't need too much more than what you've covered in the A-Level for STEP. Off the top of my head, additional things to know are Maclaurin series, proof by induction, sketching the graphs of rational functions and probably a few others that you'll easily spot if you look at their syllabus for STEP I.


Thanks for that. :smile:

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