What is the integral ∫x(t)dt

classical-mechanics physics euler-lagrange-equation

$m\ddot{x}=-kx\implies \int m(\dot{x}\ddot{x})dt=\int -kx\dot{x}dt$

$m\dot{x}^2/2=-kx^2/2+C$

Worth noting $C$ is a constant of motion, the total energy.

$\dot{x}=\sqrt{2C-kx^2}/\sqrt{m}$

$\frac{dx}{\sqrt{(2C-kx^2)/m}}=dt$

Integrate both sides, the left using trig substitution. Can you take it from there?

Related

What´s the value of the area of the triangle below?
Evaluate this limit $(\infty/\infty)$
Normalisation using Softmax- What advantage does exponential provide
All solutions to $1/a+1/b=1/c$?
Where can I find a proof that a $C^*$-algebra is nuclear iff it has completely positive approximation property?
GMAT related question: If the standard deviation of $a,b$ and $c$ is $d$. What is the standard deviation of $105\%$ of $a$ ,$105b/100$ and $1.05c$?
Relation between incentre of a triangle and a circle touching its two sides and circumcircle
Locus of centre of circle
How degree of freedom of lines in 3D space is 4?
Does every compact set in a normed space with a non-trivial interior has 2 path connected points in the boundary?
If a set $E\subset [a,b]$ has possitive measure, then $x-y\in \mathbb{R\setminus Q}$
Finding the minimum value of $|z-3+i|$ given $z$ satisfies $\arg{(z-2i)}=\frac{\pi}{6}$