> Please could someone explain how and why the limits have been applied on line 5?

From the problem statement in the second picture:

> When S is at a height of 3R above the surface of the Earth, the speed of S is √(2gR)

The surface of the Earth is x=R, so 3R above the surface is x=4R. So:

x = 4R => v = √(2gR)

You're asked to find the speed at which it hits the surface of the earth, i.e. the value V such that v=V when x=R.

x = R => v = V

> I do not understand why the expression has been integrated without adding a plus c, could you not achieve the same result by adding plus c and then solving for the particular solution?

You could.

If the indefinite integral (antiderivative) is

∫ f(x) dx = F(x) + c

then the definite integral is

∫[a,b] f(x) dx = F(b)-F(a)

So if

∫ f(x) dx = ∫ g(y) dy

=> F(x) = G(y) + c

(x=x0 => y=y0) => F(x0) = G(y0) + c => c = F(x0) - G(y0)

(x=x1 => y=y1) => F(x1) = G(y1) + c = G(y1) + F(x0) - G(y0) = F(x0) + G(y1)-G(y0)

Using the definite integral:

∫[x0,x1] f(x) dx = ∫[y0,y1] g(y) dy

=> F(x1)-F(x0) = G(y1)-G(y0)

=> F(x1) = F(x0) + G(y1)-G(y0)

However you express it, the calculations are the same.