# p5 integration

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#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
0
15 years ago
#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.
0
15 years ago
#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
0
15 years ago
#4
do they come from past papers?
0
15 years ago
#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.
0
15 years ago
#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?
0
15 years ago
#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.
0
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