I would like to know how do we solve d2x/dt2 = k' where k' is a constant i.e the task is to find x as a function of time ? One way to approach this is to rewrite it as vdv/dx = k' where v=dx/dt and first find find v as a function of x and then rewrite v as dx/dt and then find x as a function of time . I will present my attempt . vdv/dx = k' vdv= k'dx ∫vdv= ∫k'dx v2 = 2k'x + 2C' where C' is a constant. v=√(kx+C) Now,v=dx/dt dx/√(kx+C) =dt ∫dx/√(kx+C) =∫dt x = (αt+β)2 ,where α and β are some constants. Now ,I would like to know how do we solve the equation d2x/dt2 = k' by direct integration. ∫(d2x/dt2)dt = ∫k'dt How to integrate the left hand side as it is a second order derivative ?