# Math Help - Convergence of the integral

1. ## Convergence of the integral

I need help please how to solve this exercise if the integral is convergence or not

1/(x^(1/3) + x^(3/2)) dx
0

thanks a lot!

2. Originally Posted by tukilala
I need help please how to solve this exercise if the integral is convergence or not

1/(x^(1/3) + x^(3/2)) dx
0

thanks a lot!
Note that $\frac{1}{\sqrt[3]{x}+\sqrt[2]{x^3}}\leqslant\frac{1}{\sqrt[2]{x^3}}$. Now, what can you say about the infinity end of the integral? Similarly $\frac{1}{\sqrt[3]{x}+\sqrt[2]{x^3}}\leqslant\frac{1}{\sqrt[3]{x}}$, so what can you say about the zero side?

3. $\int_0^{\infty} \frac{dx}{ x^{1/3} + x^{3/2} }$

$= \int_0^1 \frac{dx}{ x^{1/3} + x^{3/2} } + \int_1^{\infty} \frac{dx}{ x^{1/3} + x^{3/2} }$

$\leq \int_0^1 \frac{dx}{ x^{1/3} } + \int_1^{\infty} \frac{dx}{ x^{3/2}}$