Question

Определенный интеграл, помогите, пожалуйста!!!!

Answer 1

$\displaystyle\int\limits^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}g(f(x))dx=\int\limits^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}g(cos(2x))dx=\int\limits^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}(2cos^2(2x)-1)dx=\\=\int\limits^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}(cos^2(2x)-sin^2(2x))dx=\int\limits^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}cos(4x)dx=\frac{1}{4}\int\limits^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}cos(4x)d(4x)=$

$\displaystyle=\frac{1}{4}sin(4x)|^{\displaystyle\frac{5\pi}{24}}_{\displaystyle0}=\frac{1}{4}(sin\frac{5\pi}{6}-sin0)=\frac{1}{4}*\frac{1}{2}=\frac{1}{8}=0.125$