ARGH!!!! Someone please help me with this integral! *Update* solved!

[RaynorWolfcastle]

I've been trying to figure out how to do this integral for 1.5 hours with no luck so far, can someone please help me compute this integral:

integral: Sqrt((t+1)^-4 + (t+1)^-2)dt from 0 to 2

Trying to solve this integral is like repeatedly slamming your head into the wall... Unless I missed an obvious solution

Any help would, as usual, be appreviated

-Ice

 

[punkrawket]

it's 11:20 at night.....

i'm not gonna think about anything like that till noon tomorrow

 

[minendo]

Start by letting x=(t+1)^-2 (I think)

 

[RaynorWolfcastle]

<< Start by letting x=(t+1)^-2 (I think) >>



I've tried that substitution but it leads nowhere as far as I can tell

-Ice

 

[LoneNutcase]

dt Sqrt(1/(1+t)^4+1/(1+t)^2) x

I think. I don't know. I'm an idiot.

 

[RaynorWolfcastle]

<< dt Sqrt(1/(1+t)^4+1/(1+t)^2) x >>



huh? how did the dt end up there and where did that x come from?

-Ice

 

[minendo]

<<

<< Start by letting x=(t+1)^-2 (I think) >>



I've tried that substitution but it leads nowhere as far as I can tell

-Ice >>


then could you complete the square? or wait maybe this is one of those d@mn trig substitutions.

 

[kulki]

<<

<< dt Sqrt(1/(1+t)^4+1/(1+t)^2) dt >>



huh? how did the dt end up there and where did that x come from?

-Ice >>


How about setting t -> tan^2 (theta)

 

[silverpig]

Maple to the rescue... uhhh, it's pretty ugly man. I'll just give the evaluated definite integral, as the indefinite integral is messy (arcsinh(x+1) etc...).

The final answer is -(1/3)sqrt(10)-ln(-3+sqrt(10))+sqrt(2)-ln(1+sqrt(2)).

 

[RaynorWolfcastle]

<< Maple to the rescue... uhhh, it's pretty ugly man. I'll just give the evaluated definite integral, as the indefinite integral is messy (arcsinh(x+1) etc...).

The final answer is -(1/3)sqrt(10)-ln(-3+sqrt(10))+sqrt(2)-ln(1+sqrt(2)). >>



That's just dirty, but how the heck am I supposed to show how I got there?

Trig substitution is a dead end BTW , it can't get rid of the root with trig. sub. From the looks of it, multiple integration by parts may work

-Ice

 

[silverpig]

Try subbing u=(t+1)

you get sqrt(u^-4 + u^2)

get them over a common denominator:

sqrt(u^2/u^6 + u^4/u^6)

simplify to:

sqrt([u^2+u^4/u^6])

take the root of the bottom:

sqrt(u^2+u^4)/u^3

factor out a u^2 and bring it out...

u sqrt(1+u^2)/u^3

cancel the u

sqrt(1+u^2)/u^2

maybe you can see something from there...

 

[silverpig]

Hmm, even when I put that into maple, it still give a thing with arcsinh and stuff (ugly)... and I'm sure that if you subbed t+1 back in, it'd give you what I already did...

 

[RaynorWolfcastle]

I've tried many substitutions including that one, I can't get it to work out nicely. I avoided trig subs because it just makes it messier.... damn arc length of parametric equations :|

I give up, if I lose points for it, I'll lose points

-Ice

 

[silverpig]

Oh oh oh...

now sub sin theta in for u...


sqrt(1+sin^2 theta)/sin^2 theta

equals

cos(theta)/sin^2(theta)

equals

cot(theta)csc(theta)

there's gotta be some kind of integral table that can give you that... then re-sub...

 

[silverpig]

oops that's wrong... forgot the d(theta) or whatever

 

[RaynorWolfcastle]

d(theta) is the problem, it creates more trouble than it solves.

thank you everyone, for the help anyways

-Ice

 

[RaynorWolfcastle]

If anyone's interested, I ended up solving it...
I had to use trig sub twice, partial fractions twice, and straight substitution once.

What a pain in the @ss!

-Ice

 

[huanaku]

this site: The Integrator can be helpful with integrals