Using knowledge of derivatives, what is $f(x)$ if $f'(x) = 4x + e^x$ with initial condition $f(0)=2$? How to solve this with knowledge of derivatives? Is it asking to solve this without using anti-derivatives?

Integrate both sides to get $f(x)$ and the equation becomes 
$$f(x) = 2x^2 + e^x + c.$$
Put $x=0$ and $f(x=0)=2$ to get $c=1$. Substitute the value of $c$ and get 
$$f(x)=2x^2+e^x+1.$$
