1. Given data

• Self inductance: $L = 2.0\,\text{H}$
• Current: $I = 2\sin(t^2)$ A

We need the energy spent during the interval in which current changes from $0$ to $2$ A.

2. Energy stored in an inductor

The magnetic energy stored in an inductor carrying current $I$ is

$U = \frac{1}{2}LI^2$

So, the energy spent in increasing current from $I_1$ to $I_2$ is

$\Delta U = \frac{1}{2}L\left(I_2^2 - I_1^2\right)$

3. Initial and final currents

• Initial current: $I_1 = 0$
• Final current: $I_2 = 2\,\text{A}$

Thus,

$\Delta U = \frac{1}{2}(2)\left(2^2 - 0^2\right)$

$\Delta U = 1 \cdot 4 = 4\,\text{J}$

4. Why the given time dependence is not needed

Although current is given as $I = 2\sin(t^2)$, the energy stored in an ideal inductor depends only on the current value, not on how fast or in what manner it reaches that value.

Hence only the change from $0$ A to $2$ A matters.

5. Final answer

$\boxed{4}$

So, the amount of energy spent is $4\,\text{J}$.