Derive second equation of motion by calculus method.

Hint: Equations of motion in kinematics describe the basic concept of the motion of an object such as the position, velocity or the acceleration of an object at various times. There are three equations of motion and these three equations of motion govern the motion of an object in 1D, 2D and 3D respectively. The equations of motion can be derived using the following methods: Derivation of equations of motion by Simple Algebraic Method, Derivation of equations of Motion by Graphical Method and Derivation equations of Motion by Calculus Method. Here we are asked to derive a second equation of motion by calculus method.

Complete step by step answer:
In physics, equations of motion are defined as equations that describe the behaviour of a physical system in terms of its motion as a function of time. There are three equations of motion that can be used to derive components such as displacement (s), initial velocity, final velocity, time (t) and acceleration (a).

The following are the three equations of motion:
First Equation of Motion: $v=u+at$
Second Equation of Motion: $s=ut+\dfrac{1}{2}a{{t}^{2}}$
Third Equation of Motion: ${{v}_{2}}={{u}_{2}}+2as$

In the next section, the second equation of motion is derived by calculus method in a simple and easy manner to understand. Now, we will derive of Second Equation of Motion by Calculus Method:
Velocity is known as the rate of change of displacement.
Mathematically, it is expressed as
$v=\dfrac{ds}{dt}$
After rearranging the equation, we get
$ds=vdt$
Substituting the first equation of motion in the above equation, we get
$ds=\left( u+at \right)dt$
$\Rightarrow ds=\left( u+at \right)dt=\left( udt+atdt \right)$
On further simplification, the equation becomes:
$\int\limits_{0}^{s}{ds}=\int\limits_{0}^{t}{udt}+\int\limits_{0}^{t}{atdt}$
$\therefore s=ut+\dfrac{1}{2}a{{t}^{2}}$

Note: We can also derive the second equation of motion by algebraic method and graphical method. For deriving second equation of motion by algebraic method we will need the formula $Displacement=\dfrac{\left( Initial\,Velocity+Final\,Velocity \right)}{2}\times time$ and for deriving second equation of motion by graphical method we will need the area under the graph.