1. ## Simple antiderrivative

Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

Thanks!

2. Originally Posted by Possible actuary
Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

Thanks!
this is an improper integral, so we have to use limits when evaluating it. for the anti derivative, we use substitution.

int{0-->infinity} [1/sqrt(x + 1)]dx

= int{0-->infinity} [(x + 1)^(-1/2)]dx
let u = x + 1
=> du = dx

so our integral becomes:

int{u^(-1/2)}du
= 2u^(1/2) + C
= 2(x + 1)^(1/2) + C

now to evaluate between 0 and infinity, we proceed like this

int{0-->infinity} [(x + 1)^(-1/2)]dx = lim{N-->infinity} int{0-->N} [(x + 1)^(-1/2)]dx

= lim{N-->infinity} 2(x + 1)^(1/2) evaluate between N and 0

= lim{N-->infinity} 2(N + 1)^(1/2) - 2(0 + 1)^(1/2)
= lim{N-->infinity} 2(N + 1)^(1/2) - 2
= infinity

3. Originally Posted by Possible actuary
Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

Thanks!
What Jhevon said, but that it diverges can be deduced by observing that
1/sqrt(x+1) decreases more slowly than 1/(x+1) the integral over (0,r) of
which diverges as r -> infty.

RonL

4. Originally Posted by Possible actuary
Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

Thanks!
INT 1/sqrt(x+1) dx = 2*(x+1)^(1/2)+C

Now evaluate the integral at limits of 1 and N to get,

2*(N+1)^(1/2)-2*(1+1)^(1/2)

Now take N --> oo

5. Originally Posted by ThePerfectHacker
INT 1/sqrt(x+1) dx = 2*(x+1)^(1/2)+C

Now evaluate the integral at limits of 1 and N to get,

2*(N+1)^(1/2)-2*(1+1)^(1/2)

Now take N --> oo
Oh yeah, that's right. the limits of integration were from 1 to infinity. i did for 0 to infinity. the answer is pretty much the same though, except for the third to last line, you plug in 1 instead of 0. the answer remains the same, it diverges to infinity.

= lim{N-->infinity} 2(N + 1)^(1/2) - 2(1 + 1)^(1/2)
= lim{N-->infinity} 2(N + 1)^(1/2) - 2sqrt(2)
= infinity

dang! i've got to learn to read again! i've been misreading posts all week. i guess it's because i'm hardly getting any sleep these days, i'm constantly tired--but maybe that's a sorry excuse

6. Thanks! Could not get the simple integration going.

7. Originally Posted by Possible actuary
Thanks! Could not get the simple integration going.
that's what we're here for