A force $$\vec{F} = (2 + 3x)\hat{i}$$ acts on a particle in the $$x$$ direction where $$F$$ is in Newton and $$x$$ is in meter. The work done by this force during a displacement from $$x = 0$$ to $$x = 4$$ m is _______ J.

The work done by a variable force is calculated by integration:

$$W = \int_{x_1}^{x_2} F \, dx$$

We are given that $$F = (2 + 3x)$$ N, displacement from $$x = 0$$ to $$x = 4$$ m

$$W = \int_0^4 (2 + 3x) \, dx$$

$$W = \left[2x + \frac{3x^2}{2}\right]_0^4$$

$$W = \left(2(4) + \frac{3(16)}{2}\right) - 0$$

$$W = 8 + 24 = 32 \text{ J}$$

The work done is $$32$$ J.