So i have a function $$F: [0,\frac{\pi}{4}]\rightarrow \mathbb{R}, F(x)=\tan^n(x), n\in\mathbb{N}$$
So from that i get:
$$V=\pi\int_0^{\frac{\pi}{4}} \tan^{2n} dx $$
I know that i have to look for a sequence when it comes to integrals for different $n$
But i get this(need to put bounds in it):
$n=1: 1-\frac{\pi}{4}$ $n=2: \frac{\pi}{4}-\frac{2}{3}$ $n=3: -\frac{\pi}{4} - \frac{13}{15}$ $n=4: \frac{\pi}{4}-\frac{76}{105}$
So i see a patern when it comes to $\frac{\pi}{4}$ But i don't see a pattern with the second fractions.
I used wolphram whether i got the right answer for those $n$, so it is correct, but how can i present the solution for diff. n and also a general one.