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RoganSarine
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[Solved] Improper Integral Integration
Sorry, don't know how to use the latex stuff for integrals :P
Integrate the following from 0 to infinity: 1/(sqrt[x]*(1+x))
Integrate 0 to 1: 1/(sqrt[x]*(1+x))
Integrate from 1 to infinity: 1/(sqrt[x]*(1+x))
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)) [tex]\geq[/tex] 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.
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)) [tex]\geq[/tex] 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.
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