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| Blob |
 Posted: Sun Apr 27, 2008 7:04 am Post subject: Partialintegration help |
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Forum Freshman
Joined: 15 Nov 2006
Posts: 11
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I'm having some math problems here which I don't know how to solve, I'm really clueless.
I don't have any program that can display integrals and stuff like that, maybe a tip on any? Also I'm not native English.
1)
Find the primitive function for e^(x) * cos (x)
2)
Find the primitive to x / ( cos(x) ^2 )
3)
Find the primitive to e^( sqrt(x) ) with 1 as higher boundary and 0 as lower.
Hint: substitute with t = sqrt(x)
Any help is really appreciated.
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| serpicojr |
| Posted: Sun Apr 27, 2008 7:29 am Post subject: |
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Forum Ph.D.
Joined: 17 Jul 2007
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Well, we don't mind helping out with homework, but we'll only help if you have made an honest effort yourself and show us what you have tried.
Having a program which can integrate things for you is a crutch that will only prevent you from learning to trust your own abilities at integration. So look at it as a blessing that you haven't been spoiled by such.
So what have you done so far? |
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| Blob |
| Posted: Sun Apr 27, 2008 7:46 am Post subject: |
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Great!
I wasn't thinking of a program to solve them for me, only something to display the integrals. Kinda like:
http://upload.wikimedia.org/math/9/1/3/9139899de25f8fe8281820ac8648a1f7.png - borrowed from wiki.
I type int()dx for the integration for now
1)
int( e^(x) * cos(x) )dx = e^(x) * sin(x) - int( e^(x) * sin(x) )dx = e^(x) * sin(x) - ( - e^(x) * cos(x) - int( - e^(x) * cos(x) )dx = e^(x) * (sin(x) + cos(x) ) - int( e^(x) * cos(x) )dx.
The last is where I'm sure I went wrong, cause that would mean it's endless going between 0 and 1 or something...
At first I did something like this:
int( e^(x) * cos(x) )dx = e^(x) * sin(x) - int( e^(x) * sin(x) )dx
After you could break out e^(x) from int( e^(x) * sin(x) )dx and get the answer e^(x) * ( sin(x) + cos(x) ) + C.
This is almost correct, only that the answer should be 1/2 times that... this is how far I came with this.
2)
int( x / ( cos(x)^2 )dx = int( x * 1 / ( cos(x)^2 ) )dx. Here I need to find the integration of 1 / ( cos(x)^2 ) which can't figure out how to do really. First I tried to substitute cos(x)^2 with t and got ln (t) but the answer surely can't be ln( cos(x)^2 ) to that.
3)
Here I'm just stuck trying to substitute the sqrt(x). If t = sqrt(x) then dt/dx = 1 / ( 2sqrt(x) ). But how you get e^( sqrt(x) ) * dx to match there I'm lost.
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| JaneBennet |
| Posted: Sun Apr 27, 2008 8:19 am Post subject: |
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| Blob wrote: |
| Here I need to find the integration of 1 / ( cos(x)^2 ) which can't figure out how to do really. |
1 ⁄ cos2(x) = sec2(x)
Does that ring a bell? 
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| Blob |
| Posted: Sun Apr 27, 2008 8:20 am Post subject: |
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Forum Freshman
Joined: 15 Nov 2006
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| JaneBennet wrote: |
| Blob wrote: |
| Here I need to find the integration of 1 / ( cos(x)^2 ) which can't figure out how to do really. |
1 ⁄ cos2(x) = sec2(x)
Does that ring a bell? |
Sec? err not really 
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| JaneBennet |
| Posted: Sun Apr 27, 2008 8:31 am Post subject: |
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sec = secant = 1 ⁄ cosine
Hint: Differentiate tan(x). What do you get?
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| Blob |
| Posted: Sun Apr 27, 2008 9:15 am Post subject: |
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I get 1 + tan2(x). I think I know where you're going; that it's somehow connected but I can't make the connection =/
Edit: Oh man it suddenly dawned to me that it's the same as 1 / cos2(x), which means that...
int( x / cos2(x )dx = x * tan(x) - int( tan(x) )dx
This however brings up a new problem: finding primitive to tan(x).
int( sin(x) * 1 / cos(x) )dx = -cox(x) * 1 / cos(x) - int( -cos(x) * - 1 / sin2(x) )dx = -1 + int( cos(x) / sin2(x) )dx
Which leaves me stuck yet again, damnet =/
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| Blob |
| Posted: Sun Apr 27, 2008 1:40 pm Post subject: |
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I found the solutions the the problems now =)
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| JaneBennet |
| Posted: Mon Apr 28, 2008 3:32 am Post subject: |
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| Blob wrote: |
| This however brings up a new problem: finding primitive to tan(x). |
tan(x) = sin(x) ⁄ cos(x) = −[−sin(x) ⁄ cos(x)]
and the expression in the square brackets is of the form f′(x) ⁄ f(x). You should know how to integrate expressions of the form f′(x) ⁄ f(x).
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