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 $\displaystyle \frac{1}{\sqrt[3]{x}+\sqrt[2]{x^3}}\leqslant\frac{1}{\sqrt[2]{x^3}}$. Now, what can you say about the infinity end of the integral? Similarly $\displaystyle \frac{1}{\sqrt[3]{x}+\sqrt[2]{x^3}}\leqslant\frac{1}{\sqrt[3]{x}}$, so what can you say about the zero side?
$\displaystyle \int_0^{\infty} \frac{dx}{ x^{1/3} + x^{3/2} } $
$\displaystyle = \int_0^1 \frac{dx}{ x^{1/3} + x^{3/2} } + \int_1^{\infty} \frac{dx}{ x^{1/3} + x^{3/2} } $
$\displaystyle \leq \int_0^1 \frac{dx}{ x^{1/3} } + \int_1^{\infty} \frac{dx}{ x^{3/2}} $