# Partialintegration help

• April 27th, 2008, 09:04 AM
Blob
Partialintegration help
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.
• April 27th, 2008, 09:29 AM
serpicojr
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?
• April 27th, 2008, 09:46 AM
Blob
Great!

I wasn't thinking of a program to solve them for me, only something to display the integrals. Kinda like:

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.
• April 27th, 2008, 10:19 AM
JaneBennet
Quote:

Originally Posted by Blob
Here I need to find the integration of 1 / ( cos(x)^2 ) which can't figure out how to do really.

1 ⁄ cos<sup>2</sup>(x) = sec<sup>2</sup>(x)

Does that ring a bell? ;)
• April 27th, 2008, 10:20 AM
Blob
Quote:

Originally Posted by JaneBennet
Quote:

Originally Posted by Blob
Here I need to find the integration of 1 / ( cos(x)^2 ) which can't figure out how to do really.

1 ⁄ cos<sup>2</sup>(x) = sec<sup>2</sup>(x)

Does that ring a bell? ;)

Sec? err not really :(
• April 27th, 2008, 10:31 AM
JaneBennet
sec = secant = 1 ⁄ cosine

Hint: Differentiate tan(x). What do you get?
• April 27th, 2008, 11:15 AM
Blob
I get 1 + tan<sup>2</sup>(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 / cos<sup>2</sup>(x), which means that...

int( x / cos<sup>2</sup>(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 / sin<sup>2</sup>(x) )dx = -1 + int( cos(x) / sin<sup>2</sup>(x) )dx

Which leaves me stuck yet again, damnet =/
• April 27th, 2008, 03:40 PM
Blob
I found the solutions the the problems now =)
• April 28th, 2008, 05:32 AM
JaneBennet
Quote:

Originally Posted by Blob
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). ;)