[tex]\displaystyle\bf\\f(x)=5Sin\frac{x}{2} +2Cos3x\\\\\\F(x)=5\cdot\Big(-2Cos\frac{x}{2}\Big)+2\cdot\frac{1}{3} Sin3x+C\\\\\\F(x)=-10Cos\frac{x}{2}+\frac{2}{3} Sin3x+C\\\\\\\Big(\frac{\pi }{2} \ ; \ -\frac{2}{3} \Big) \ \ \ \Rightarrow \ \ \ x=\frac{\pi }{2} \ \ ; \ \ F(x)=-\frac{2}{3} \\\\\\-\frac{2}{3} =-10Cos\frac{\pi }{4} +\frac{2}{3} Sin\frac{3\pi }{2} +C\\\\\\-\frac{2}{3} =-10\cdot\frac{\sqrt{2} }{2} +\frac{2}{3} \cdot(-1)+C\\\\\\-\frac{2}{3}=-5\sqrt{2} -\frac{2}{3} +C[/tex]

[tex]\displaystyle\bf\\C=5\sqrt{2} \\\\\\Otvet \ : \ F(x)=-10Cos\frac{x}{2} +\frac{2}{3} Sin3x+5\sqrt{2}[/tex]