If
\[ f(x) = \begin{cases} \sin x & \text{for } x < 0 \\ 3x^{2} & \text{for } x \geq 0 \end{cases} \]
find \( \int_{-\pi/2}^{\pi/2} f(x) \, dx \).
calculus-idefinite-integralpiecewise-functionintegrationtrigonometric-functionspolynomial-functionsfundamental-theorem-of-calculusadditivity-of-integralsevaluate-integralsinecosine
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