$\Rightarrow \int_{-\pi}^{t} f(x) d x+\int_{-\pi}^{t} x d x=\pi^{2}-t^{2}$
$\Rightarrow \int_{-\pi}^{t} f(x) d x+\left(\frac{t^{2}}{2}-\frac{\pi^{2}}{2}\right)=\pi^{2}-t^{2}$
$\Rightarrow \int_{-\pi}^{t} f(x) d x=\frac{3}{2}\left(\pi^{2}-t^{2}\right)$
differentiating with respect to $t$
$\frac{d}{d t}\left[\int_{-\pi}^{t} f(x) d x\right]=\frac{3}{2} \frac{d}{d t}\left(\pi^{2}-t^{2}\right)$
$f(t) \cdot \frac{d t}{d t}-f(-\pi) \frac{d}{d t}(-\pi)=-3 t$
$f(t)=-3 t$
$f\left(-\frac{\pi}{3}\right)=-3\left(-\frac{\pi}{3}\right)=\pi$
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