Can You Solve This Challenging Improper Integral?

[RoganSarine]

[Solved] Improper Integral Integration

Sorry, don't know how to use the latex stuff for integrals :P

Homework Statement

Integrate the following from 0 to infinity: 1/(sqrt[x]*(1+x))

Homework Equations

Integrate 0 to 1: 1/(sqrt[x]*(1+x))
Integrate from 1 to infinity: 1/(sqrt[x]*(1+x))

The Attempt at a Solution

Integrate from 1 to infinity: 1/(sqrt[x]*(1+x))
This is convergent because you can tell as it goes to infinity, it will approach 0.

So, I should (i think) be able to find the integral of this, but... I can't use improper fractions because that square root messes everything up (atleast, from my understanding how improper fractions work... When I try to break it up, the square root always ends up negative).

Integrate 0 to 1: 1/(sqrt[x]*(1+x))
This shoots off to infinity as the function approaches zero, so...

If b = 0
lim x -> b+ Integrate b to 1: 1/(sqrt[x]*(1+x))

If x gets really close to zero, I can assume
1/(sqrt[x]*(1+x)) ~ 1/(sqrt[x])

Therefore, by using comparison tests,

1/(sqrt[x]*(1+x)) \geq 1/(sqrt[x])

Since I know that (1/x^p) is convergent if p<1 for any bounds between 0-1.

Basically, I know the theory... I just got no idea how to solve the rest of it.

 

[Bohrok]

Try the substitution u = √x