Question

How can I prove that this converges?

Answer 1

\[\int\limits_{0}^{\frac{ \pi }{ 2 }}\tan ^{2}(2x)dx\]
Edit: It doesn't converge, it diverge.

Answer 2

I seperated into 0 to pi/4 and to pi/4 to pi/2 because the "blow-up" point is at pi/4

Answer 3

\[\int\limits_{0}^{\frac{ \pi }{ 4 }}+\int\limits_{ \frac{ \pi }{ 4 }}^{\frac{ \pi }{ 2 }}\] it goes to + infinity but do I just write that? Is there anything else I could calculate to prove it. limit comparison or something?

Answer 4

Please note that you have to calculate these quantities:
\[\lim _{\sigma \rightarrow \pi/4-}\int\limits_{0}^{\sigma} [\tan(2x)]^{2}dx\]
and:
\[\lim _{\delta \rightarrow \pi/4+}\int\limits\limits_{\delta}^{\pi/2} [\tan(2x)]^{2}dx \]

Answer 5

furthermore, keep in mind that:
\[\int\limits [\tan (2x)]^{2}dx=\frac{ \tan(2x)-(2x) }{ 2 }+c\]
where c is a real arbitrary constant