A particle of mass m travels on a straight line, with its velocity...

According to the work-energy theorem, the total work done by all forces is equal to the change in kinetic energy of the particle.

$W = K_f - K_i$

Given the velocity equation $v = a\sqrt{x}$.

At $x = 0$, the initial velocity is $v_i = 0$, so the initial kinetic energy is $K_i = 0$.

At $x = d$, the final velocity is $v_f = a\sqrt{d}$.

The final kinetic energy is $K_f = \dfrac{1}{2} m v_f^2 = \dfrac{1}{2} m (a\sqrt{d})^2 = \dfrac{m a^2 d}{2}$.

Therefore, the total work done is $W = \dfrac{m a^2 d}{2} - 0 = \dfrac{m a^2 d}{2}$.