Does this integral converge?

Question

∫₀^{∞ 4}√x / (√x + x³) dx

I tried using Direct Comparison Test, but could not come up with a function g(x) if f(x) is the above function.

Answer 1

First simplify the function

f(x) = ⁴√x/(√x + x³) = 1/(x^1/4 + x^11/4) = 1/(x^1/4(1 + x^5/2))

Now derive two inequalities.

On the interval (0, ∞)

x^5/2 > 0 therefore

1 + x^5/2 > 1 therefore

x^1/4(1 + x^5/2) ≥ x^1/4 therefore

f(x) ≤ 1/x^1/4 (1)

On the interval (1, ∞)

x^1/4 ≥ 1 therefore

x^1/4(1 + x^5/2) ≥ 1 + x^5/2 ≥ x^5/2 therefore

f(x) ≤ 1/x^5/2 (2)

Rewrite the given integral from 0 to ∞ as a sum of two integrals:

from 0 to 1 and from 1 to ∞.

For the integral 0 to 1 you can use (1) because

∫1/x^1/4 = 4x^3/4/3 + C

For the integral from 1 to ∞ you can use (2) because

∫1/x^5/2 = -2/3x^3/2 + C