Question

Найти интеграл от П/24 до 0 tg^2*4x

Answer 1

$\int\limits^{\frac{\pi }{24}}_0\, tg^24x\, dx=\int\limits_0^{\frac{\pi }{24}}\, (\frac{1}{cos^24x}-1)\, dx=(\frac{1}{4}\, tg4x-x)\Big |_0^{\frac{\pi}{24}}=\frac{1}{4}\cdot tg\frac{\pi}{6}-\frac{\pi}{24}=\\\\=\frac{1}{4}\cdot \frac{\sqrt3}{3}-\frac{\pi}{24}=\frac{\sqrt3}{12}-\frac{\pi}{24}\\\\\\\star \; \; 1+tg^2a=\frac{1}{cos^2a}\; \; \; \Rightarrow \; \; \; tg^2a=\frac{1}{cos^2a}-1\; \; \star$