# Improper Integral Integration

1. Feb 9, 2010

### RoganSarine

[Solved] Improper Integral Integration

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

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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.

Last edited: Feb 9, 2010
2. Feb 9, 2010

### Bohrok

Try the substitution u = √x