Definite Integration

Use properties of definite integrals (specifically the symmetry property for even and odd functions) to evaluate $\int_{-2}^{2} (x^4 - 2x^2 + 5) \, dx$.

The velocity of a particle moving along a line is $v(t)=t^3-3t^2+2t+1$ m/s. Use properties of definite integrals (such as symmetry) to find the net displacement from $t=-2$ s to $t=2$ s.

In a thermodynamics experiment, the rate of heat transfer across a boundary is given by the function $f(x) = (x^3 + 2x^2 + x + 1)e^{x^2}$ (in kW) over the interval $x \in [-2, 2]$ (in meters). Determine the total heat transferred over this interval, i.e., evaluate $\int_{-2}^{2} (x^3 + 2x^2 + x + 1)e^{x^2} \, dx$.