Integral calculus
      The indefinite integral applications
         Kinetic equation of free fall
         Exponential growth and decay (increase and decrease)
The indefinite integral applications
In physics, physical laws are described by mathematical formulas.
Kinetic equation of free fall
For example kinetic quantity velocity v, defined as the rate at which position s changes with respect to time t,
represents an antiderivative or primitive function of v (t), hence
Since, at t = t0, the formula for displacement gives  s(t0) = 0 then, the constant t0 denotes the point in time when starts measurement of the displacement.
Thus for example, the velocity of an object in free fall motion
v = gt,
where g = 9,81 m/s/s is the acceleration of free fall (the acceleration due to gravity), therefore the displacement
 
what can be checked over by differentiating the right-hand side with respect to t to obtain the integrand gt
Exponential growth and decay (increase and decrease)
A quantity that increases or decreases with respect to time t, or other argument, such that the rate of change by which it happens at an instant t is proportional by the value of the quantity at that instant, then it changes by the exponential law.
Let x (t) denotes the value of a quantity at a time t, the rate dx/dt, at which the quantity changes at an instant t is proportional by the value of the quantity at that instant, with constant of proportionality k then, x (t) satisfies the differential equation of the exponential function
where the sign of constant of proportionality k depends on whether x(t) increases or decreases with t.
To solve the above differential equation we rewrite it by multiplying by dt and dividing by x(t) to separate variables so that each side of the equation can be integrated with respect to each variable,
By setting t = 0 to both sides of the solution gives | x(0) | = C  thus, the constant of integration defines the initial value of the quantity | x(t) |.
Therefore, the general solution of the differential equation    
dx/dt = k x (t)
is represented by the exponential function multiplied by an arbitrary constant x (0),
x (t) = x (0) ekt 
    or      y (t) = C ekt    is a solution to the differential equation    dy/dt = k y.
Recall that the graph of the exponential function  f (x) = ex  and its derivative coincide, i.e.,
 y' = dy/dx = ex = y
so that the sub tangent (projection of the segment of the tangent onto the x-axis) drawn at any point x,
 st = y / y'  = 1
as is shown in the above figure.
Therefore,  for  x(0) > 0  and  k > 0, the quantity x(t) increases as t increases while, in case the coefficient of t is negative, i.e.,  k < 0 x(t) decreases, for which an example is radioactive decay. In this case the graph of the exponential function reflects around the y-axis.
Calculus contents E
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