Help with numerical integration

[Dominguez Scaramanga]

Hello there, I've not been here in a while, but I'm stuck doing this integration and wondered if some of you kind people would help

\int_0^\infty \frac{1} {(1+x)\sqrt{x}} dx

(appologies for the lack of spacing in there...)

anyways, I know that when x tends to infinity, the integral can be approximated to,


\int_0^\infty \frac{1} {(x)\sqrt{x}} dx

but I can't seem to find this identity in any of my tables anywhere...

The reason I need it is because I'm in the processes of writing some c code to analytically calculate this with a specified degree of acuracy, (am going to use the trapezium method of integration I think) so it would be nice to know if the answers I get out of it are any good or not!

thanks for you time

 

[DeadWolfe]

use rules of exponents.

1/(x(sqrtx)) = x^(-3/2)

 

[Dominguez Scaramanga]

awesome, thanks DeadWolfe :)

 

[dextercioby]

That integral (the first) can be very simply computed analytically to yield the result \pi...Heck,u can even define \pi by it

\pi=:\int_{0}^{+\infty} \frac{dx}{(1+x)\sqrt{x}}

HINT:Make the obvious substitution
\sqrt{x}=t

Daniel.

 

[Dominguez Scaramanga]

wow, even more helpful, thanks a lot

also, would I be correct in saying that the limit of the integrand as x-->infinity is pi, in that case?
or have I got completely mudled up?

 

[dextercioby]

Define

P(x)=:\int_{0}^{x} \frac{dt}{(1+t)\sqrt{t}}

Show that

P(x)=2\arctan x

Then it's easy to say

\lim_{x\rightarrow +\infty} P(x)=\pi

Not the integrand!The integrands's (inferior) limit to 0 is +\infty,while its limit to +\infty is 0 ()

Daniel.

 

[Dominguez Scaramanga]

thanks very much for your help dextercioby, it'll be most useful!

now all I've got to do is figure out how to do this with C

also, where you have,

P(x)=:\int_{0}^{x} \frac{dt}{(1+t)\sqrt{t}}

that'll yield the same result as for

F(x)=:\int_{0}^{+\infty} \frac{dx}{(1+x)\sqrt{x}}

with the substitution

\sqrt{x}=t

right?

 

[dextercioby]

Nope.The second (the one with F(x)) is a number,while the first is a function...So in the second case,the notation is incorrect...

Daniel.

 

[Dominguez Scaramanga]

ah I see, ok, so the first one, with x = +infinity, will equal pi, the same as the bottom one would (had I wrote it correctly).

 

[dextercioby]

Yes.U got it

Daniel.

 

[Dominguez Scaramanga]

finally thanks very much for your help