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
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