Assistance needed: Real Analysis

[Euroman24]

Hi

Given the function $\displaystyle (tan(x))^{-\beta}$. Determine for which values of $\displaystyle \beta \in \mathbb{R}$, where $\displaystyle x \in ]0, \frac{\pi}{2} [$

1) Can be integreated for beta = 0.

2) Can be integrated for $\displaystyle \beta = \frac{\pi}{2}$

3) Can be integrated on interval $\displaystyle ]0,\frac{\pi}{2}[$

Solution

1)

If $\displaystyle \beta = 0$ then I get $\displaystyle I_{\beta = 0} = \int_{0} ^{\frac{\pi}{2}}1 dx = \frac{\pi}{2}$

Since $\displaystyle x = \frac{\pi}{2}$ is endpoint of the interval for x, then this integral $\displaystyle I_{\beta} = \int_{0} ^{\frac{\pi}{2}} tan(x)^{-\beta} dx$ can be integrated for the value beta = 0.

Could anyone please inform me if my conclusion here is correct?

2)

For $\displaystyle \beta = {\frac{\pi}{2}}$ then that must I need to solve the integral
$\displaystyle I_{\beta = {\frac{\pi}{2}}} = \int_{0} ^{\frac{\pi}{2}} tan(x)^{-\beta} dx$ ??

In order to calculate the integral above, I have tried to construct a reduction formula for the integral, but I'm stuck here

$\displaystyle I_{\beta} = -tan(x)^{-\beta} \cdot ln(cos(x)) + (1-\beta) \cdot \int \frac{ \cdot tan(x)^{-\beta}}{(cos(x))^2} \cdot ln(cos(x)) dx$

Could anybody please give me a hint on if I'm on the right path to solve this problem?

Sincerely Yours
Euroman24