Question

Укажите первичную для функции f(x)=3 cos 3x + 1/2 × sin × x/2, график которой проходит через точку А (пи/2; -2/3)

Answer 1

$g(x)= \int\limits [{3cos(3x)+\frac{1}{2}sin(\frac{x}{2})}] \, dx =\\\\ = \int\limits {3cos(3x)} \, dx + \int\limits {\frac{1}{2}sin(\frac{x}{2})}} \, dx =\\\\ = \int\limits {cos(3x)} \, d(3x) + \int\limits {sin(\frac{x}{2})}} \, d(\frac{x}{2}) =\\\\ =sin(3x)-cos(\frac{x}{2})+C$

$g(\frac{\pi}{2})=-\frac{2}{3}\\\\ -\frac{2}{3}=sin(3*\frac{\pi}{2})-cos(\frac{1}{2}*\frac{\pi}{2})+C\\\\ -\frac{2}{3}=-1-\frac{\sqrt{2}}{2}+C\\\\ C=\frac{1}{3}+\frac{\sqrt{2}}{2}$

$g(x)=sin(3x)-cos(\frac{x}{2})+\frac{1}{3}+\frac{\sqrt{2}}{2}$