Question

drive any two equation of uniformly accelerated motion by calculus method

Answer 1

First Equation of motion

The first equation of motion is v=u+at , where v is the final velocity and u is the initial velocity of the body.

First equation of motion gives velocity acquired by body at any time t.

Now we know that acceleration

so, a=v−ut
and, at=v−u
rearranging above equation we get first equation of motion that is
v=u+at

Second Equation of motion

Second equation of motion is
s=ut+12at2
where u is initial velocity, a is uniform acceleration and s is the distance travelled by body in time t.

Second equation of motion gives distance travelled by a moving body in time t.

To obtain second equation of motion consider a body with initial velocity umoving with acceleration a for time t its final velocity at this time be v. If body covered distance s in this time t , then average velocity of the body would be

Distance travelled by the body is

From first equation of motion 
v=u+at
So putting first equation of motion in above equation we get ,
s=u+u+at2×t=(2u+at)t2=2ut+at22
Rearranging it we get
s=ut+12at2