(i)[4]
Use the substitution $u = \tan x$ to demonstrate that, for $n \neq -1$, $\int_0^{\frac{\pi}{4}} (\tan^{n+2}x + \tan^n x)\,dx = \frac{1}{n + 1}$.
(ii(a))[3]
Hence determine the exact value of $\int_0^{\frac{\pi}{4}} (\sec^4 x - \sec^2 x)\,dx$.
(ii(b))[3]
Hence determine the exact value of $\int_0^{\frac{\pi}{4}} (\tan^9 x + 5\tan^7 x + 5\tan^5 x + \tan^3 x)\,dx$.