Integration by Substitution help needed please (1 Viewer)

[blackops23]

Hi guys, I was doing an integration question:

Q. integrate 1/{(x)[root(1+x^2)]} limits: 1(<)x(<)infinity

where i used the substitution x=tanA

and ended up with having to integrate cosecA.But the problem I had was changing the integral limits.

I mean for x=1, obviously A=pi/4. But what on earth do you do for x=infinity? I mean inverse tan (infinity) =???????

Help with this inverse tan infinity thing would greatly be appreciated.

Thankyou very much.

 

[kr73114]

wouln't the other limit be pi/2 for x=infinity. soz im not sure

 

[blackops23]

wouln't the other limit be pi/2 for x=infinity. soz im not sure

please do explain why and how you think it is pi/2...

 

[stampede]

because as x approaches infinity tan inverse x approaches pi/2

 

[blackops23]

OH SHIT lol, kept thinking of the graph of tan(x) not inverse tan(x).... (

Thanks guys for the help

FOLLOW UP QUESTION:

How do you integrate csc x?

 

[stampede]

cosec x is (sinx)^-1
then do it via t method or someshit

 

[funnytomato]

OH SHIT lol, kept thinking of the graph of tan(x) not inverse tan(x).... (

Thanks guys for the help

FOLLOW UP QUESTION:

How do you integrate csc x?

there are many ways to integrate cosec x
I would just multiply it by (cosec x +cot x )/(cosec x +cot x)

then factorise -1 at the front, then it becomes a log

 

[funnytomato]

the answer would be -ln (cot x + cosec x) if i didn't make a mistake

 

[blackops23]

the answer would be -ln (cot x + cosec x) if i didn't make a mistake

ok so say that is what happens when you have integrated cscx, with limiters pi/4(<)x(<)pi/2. cot(pi/2) is undefined -- so how do i do this question?

Appreciate the help

 

[funnytomato]

ok so say that is what happens when you have integrated cscx, with limiters pi/4(<)x(<)pi/2. cot(pi/2) is undefined -- so how do i do this question?

Appreciate the help

cot(pi/2) is undefined???
isn't it 0?

 

[funnytomato]

hopefully a GRAPH is worth 1000 words
http://imgur.com/7ZORM

forgot how to show the graph , lol

 

[funnytomato]

I got ln (1+sqrt2)

 

[blackops23]

cot(pi/2) is undefined???
isn't it 0?

crap, it is... I thought tan pi/2 was undefined, so cot pi/2 was also undefined...?? like on my calculator 1/tan(pi/2) is undefined, yet the graph of cot x says otherwise?

What exactly is going on?

 

[stampede]

crap, it is... I thought tan pi/2 was undefined, so cot pi/2 was also undefined...?? like on my calculator 1/tan(pi/2) is undefined, yet the graph of cot x says otherwise? What exactly is going on?

tan pi/2 is infinity basically or approaches it
thus 1/tan(pi/2) is like saying 1/10000000000000000000000000 which is pretty much going towards 0
do u understand

 

[blackops23]

tan pi/2 is infinity basically or approaches it
thus 1/tan(pi/2) is like saying 1/10000000000000000000000000 which is pretty much going towards 0
do u understand

but isnt that involving limits and such? I mean, the exact value of tan pi/2 is undefined... or is it infinity??

 

[stampede]

but isnt that involving limits and such? I mean, the exact value of tan pi/2 is undefined... or is it infinity??

mate youve just got to accept and live with it

 

[Trebla]

The strict definition of cot x is NOT 1/tan x. The definition of cot x is in fact (cos x)/(sin x) and it is only equal to 1/tan x if tan x is defined.

 

[blackops23]

The strict definition of cot x is NOT 1/tan x. The definition of cot x is in fact (cos x)/(sin x) and it is only equal to 1/tan x if tan x is defined.

aahh i see thanks trebla, and yeah i did get ln(1+root2)

 

[Drongoski]

the answer would be -ln (cot x + cosec x) if i didn't make a mistake

Answer is correct (don't forget the arbitrary constant) - easily verified by differentiating wrt x your answer - should give back cosec x

 

[funnytomato]

Answer is correct (don't forget the arbitrary constant) - easily verified by differentiating wrt x your answer - should give back cosec x

lol, this reminds me of the joke about the integral of x^2 in some Jeff Geha study guide