Is it possible to integrate acceleration?

[Madtasmo]

Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion:
Mg*sin{α} - 1.5m*x(double dot)=0

I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's possible, anyways, thanks!

 

[osilmag]

$\int \sin(x) dx = -\cos(x)$

That is the general form for integrating $\sin(x)$.

 

[osilmag]

$\int d^2x = x dx$

 

[Madtasmo]

Turns out it's possible to do so;
X(dot)=2/3*gt*sin(α)

 

[Hiero]

I had gotten to this equation of motion:
Mg*sin{α} - 1.5m*x(double dot)=0

Surely you’ve seen $\ddot x =$ constant.
To find $\dot x$ you use $\ddot x =\frac{d}{dt}\dot x$ and separate.

Another common trick worth knowing is from the chain rule:
$$ \frac{d^2x}{dt^2} =\frac{d\dot x}{dt} =\frac{dx}{dt} \frac{d\dot x}{dx} =\dot x \frac{d\dot x}{dx} $$
For example, if you have Newton’s law in the form $m\frac{d^2x}{dt^2}=F(x)$ then the chain rule makes it separable, from which we get the (1-D) work energy theorem (for point masses).

 

[nasu]

Isn't your alpha a function of time too? If alpha were constant, integrating would be trivial. But is'it? What is the physical system described by the equation?