$\int_{-2}^{2}\left(4 x^{3}-3 \sin \left(\frac{\pi}{2} x\right)\right) d x$

Step 1 ：Calculate the antiderivative of the function: \(x^4 - \frac{6}{\pi}\cos\left(\frac{\pi}{2}x\right)\)

Step 2 ：Evaluate the antiderivative at the upper limit of 2: \(16 + \frac{6}{\pi}\)

Step 3 ：Evaluate the antiderivative at the lower limit of -2: \(16 + \frac{6}{\pi}\)

Step 4 ：Subtract the lower limit evaluation from the upper limit evaluation: \(0\)

Step 5 ：The solution to the integral is \(\boxed{0}\)