Given $\frac{dy}{dx}=4\sqrt[\phantom{2}]{x}+4sin(2x),$ and $y(1)=1,$

$($a$)$ Find $y(x),$ using exact values $($for example it may involve terms like $cos(2)$ or $sin(3))$

$($b$)$ Hence find $y(2)$ exactly.

To enter you answer exactly, do not evaluate trig functions on your calculator, instead type $cos(9)$ for example.

$($c$)$ Evaluate $y(2)$ numerically, accurate to two decimal places.