Equation with derivative on the left side I have a differential equation with derivative on the left side:

$y'=x^2e^x$

My theory is, that the solution to this equation is a simple integral $\int x^2 e^x dx$.

Is my theory correct?

Thanks
Yes, you are right. You only have to calculate the antiderivative of the RHS
This is the simplest case of an ODE.

$$y'=f(x)$$ or $$dy=f(x)\,dx$$ which you integrate as

$$y=\int f(x)\,dx+c=F(x)+c.$$

Another simple case is that of the separable equation,

$$g(y)\,y'=f(x),$$

$$\int g(y)\,dy=\int f(x)\,dx+c$$

or 

$$G(y)=F(x)+c,$$

$$y=G^{-1}(F(x)+c).$$