Задача №1290
Условие
Найти интеграл \(\int\left(\cos\left(2x-\frac{\pi}{4}\right)\right)^{-2}dx\).
Решение
\[
\int\left(\cos\left(2x-\frac{\pi}{4}\right)\right)^{-2}dx
=\left[\begin{aligned}& d\left(2x-\frac{\pi}{4}\right)=2dx;\\& dx=\frac{1}{2}d\left(2x-\frac{\pi}{4}\right).\end{aligned}\right]
=\frac{1}{2}\int\frac{d\left(2x-\frac{\pi}{4}\right)}{\cos^2\left(2x-\frac{\pi}{4}\right)}
=\frac{1}{2}\tg\left(2x-\frac{\pi}{4}\right)+C
\]
Ответ:
\(\frac{1}{2}\tg\left(2x-\frac{\pi}{4}\right)+C\)