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# p5 integration watch

1. show
int (x+1)^0.5/(x+5)dx
limit 8 to 3
is
pi+2-5arctan(3/2)

use x=sint
int 1/(x+(1-x^2)^0.5)dx
limit 1 to 0
pi/4

I think i did it
but i have forgotten
2. I posted somthing on my website about the solution to your second integral, there is a nice method you can use to solve it: http://michaelmgs.f2o.org/maths_32.html.

For the first, I am assuming you mean (x+1) rather than (x+a); you need to apply the substitution u^2 = x+1.
3. (Original post by totaljj)
show
int (x+1)^0.5/(x+5)dx
limit 8 to 3
is
pi+2-5arctan(3/2)

use x=sint
int 1/(x+(1-x^2)^0.5)dx
limit 1 to 0
pi/4

I think i did it
but i have forgotten
I got pi+2-4arctan(3/2) for the first one
4. do they come from past papers?
5. (Original post by keisiuho)
do they come from past papers?
These are from exercises in the heinmann book, I think it is exercise 3C which has about 70-80 integration questions the last few of which are quite difficult.
6. (Original post by totaljj)
show
int (x+1)^0.5/(x+5)dx
limit 8 to 3
is
pi+2-5arctan(3/2)

use x=sint
int 1/(x+(1-x^2)^0.5)dx
limit 1 to 0
pi/4

I think i did it
but i have forgotten
There is another way without using the method above for the second question. But it is quite complicated
you will get:
INT cos/(sin + cos)
= INT cos*(sin - cos)/[(sin + cos)(sin - cos)]
= INT (cost sint - cos^2 t)/ (-cos2t)
= INT [0.5 sin2t - 0.5(1+cos2t)] / (-cos2t)
= INT -0.5 tan2t + 0.5 sec2t + 0.5
As you know the integrals of tan2t and sec2t, the answer can be found.
btw, can I use the results of integrals of tan2t and sec2t directly without proof in the exam?
7. (Original post by keisiuho)
There is another way without using the method above for the second question. But it is quite complicated
you will get:
INT cos/(sin + cos)
= INT cos*(sin - cos)/[(sin + cos)(sin - cos)]
= INT (cost sint - cos^2 t)/ (-cos2t)
= INT [0.5 sin2t - 0.5(1+cos2t)] / (-cos2t)
= INT -0.5 tan2t + 0.5 sec2t + 0.5
As you know the integrals of tan2t and sec2t, the answer can be found.
btw, can I use the results of integrals of tan2t and sec2t directly without proof in the exam?
Yeah, that looks like a good method. I think you can use the integrals without proof, I think they are in the formula book and can be quoted from there.

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Updated: June 19, 2004
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