Question

Вычислите определенный интеграл

Answer 1

$\int\limits^{\frac{\pi}{4}}_0\, cos^2x\, dx=\int\limits^{\frac{\pi}{4}}_0\, \dfrac{1+cos2x}{2}\, dx=\int\limits^{\frac{\pi}{4}}_0\, \Big(\dfrac{1}{2}+\dfrac{1}{2}\cdot cos2x\Big)\, dx=\\\\\\=\Big(\dfrac{1}{2}\cdot x+\dfrac{1}{2}\cdot \dfrac{1}{2}\, sin2x\Big)\Big|_0^{\frac{\pi}{4}}=\dfrac{1}{2}\cdot \dfrac{\pi}{4}+\dfrac{1}{4}\cdot (sin\dfrac{\pi}{2}-sin0)=\dfrac{\pi}{8}+\dfrac{1}{4}=\dfrac{1}{4}\cdot \Big(\dfrac{\pi}{2}+1\Big)$