Improper Integral Integration

In summary, the given improper integral from 0 to infinity of 1/(sqrt[x]*(1+x)) can be solved by using the substitution u = √x and applying comparison tests to show its convergence. The integral from 0 to 1 is divergent, while the integral from 1 to infinity is convergent.
  • #1
RoganSarine
47
0
[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)) [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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  • #2
Try the substitution u = √x
 

1. What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite or the integrand has an infinite discontinuity at one or more points within the interval of integration.

2. How do you determine if an integral is improper?

An integral is considered improper if at least one of the following conditions is met: the limits of integration are infinite, the integrand has an infinite discontinuity within the interval, or the integrand is unbounded within the interval.

3. How do you solve an improper integral?

To solve an improper integral, you must break it into smaller integrals by finding the limit as one or both of the limits of integration approach infinity or a point of discontinuity. Then, you can solve each smaller integral separately and take the limit of the overall solution as the limits of integration approach their values.

4. What are some common techniques for solving improper integrals?

Some common techniques for solving improper integrals include using the comparison test, the limit comparison test, and the convergence test. Other methods include using substitution, integration by parts, and partial fractions.

5. Why are improper integrals important in science?

Improper integrals are important in science because they allow us to solve for quantities that cannot be evaluated using standard integration techniques. They also help us understand and analyze physical phenomena that involve infinite or discontinuous behavior, such as in physics and engineering problems.

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