Determine if the integral $$\int^{2}_{0} \dfrac{\sec^2(x)}{x\sqrt{x}}dx$$ converges or diverges?

I know that it is positive on this interval so I can use the convergence/divergence theorem.

This integral has to discontinues, $x=0, x=\pi/2$, and so this makes me think that it diverges. I have to find a smaller function. I have trouble here.

How do I make the numerator $\sec^2(x)$ smaller? I can make the denominator bigger by just making it $x$, or $x^2$, but I need help with the numerator.

## Best Answer

I prefer to use this:

$$\frac{\sec^2(x)}{x^{3/2}}\ge\frac1{x^{3/2}}$$

It thus follows that

$$\int_0^2\frac1{x^{3/2}}\ dx=\frac{-2}{x^{1/2}}\bigg|_0^2$$

And we immediately conclude from there.