[tex]\int {cos2x+2sin^2x+3\sqrt{x} } \, dx =\int {cos2x } \, dx+\int {2sin^2x} \, dx+\int {3\sqrt{x} } \, dx=\\\\=\frac{1}{2}\int {cos2x } \, d(2x)+2\int {\frac{1-cos2x}{2}} \, dx+3\int {x^{\frac{1}{2}} } \, dx=\frac{1}{2}sin2x+(x-\frac{1}{2}sin2x)+3\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\\\\=x+2\sqrt{x}[/tex]