How to derive an equation of motion by integration?

For first equation of motion, the definition of acceleration can be used.

For second equation of motion you can find the area under the v-t graph to find the distance covered by the body. For this integration can be used.

a =	dv
	dt
		dv =	a dt
	v		Δt
	⌠ ⌡	dv =	⌠ ⌡	a dt
	v0		0
v − v0 =			aΔt
		v =	v0 + aΔt		[1]

 

v =			dx
			dt
dx =			v dt
	x		Δt		Δt
	⌠ ⌡	dx =	⌠ ⌡	v dt =	⌠ ⌡	(v0 + aΔt)dt
	x0		0		0
x − x0 =			v0Δt + ½ aΔt2
x =			x0 + v0Δt + ½ aΔt2				[2]