(C)
\( \int_{-3}^{3} \cot ^{-1} x d x=\int_{-3}^{3} \cot ^{-1} x \cdot 1 d x \)
\( =\left[\cot ^{-1} x \cdot x\right]_{-3}^{3}-\int_{-3}^{3} x\left(-\frac{1}{1+x 2}\right) d x \)
\( =3 \cot ^{-1} 3-(-3) \cot ^{-1}(-3)+\int_{-3}^{3} \frac{x}{1+x^{2}} d x \)
\( =3 \cot ^{-1} 3+3\left(\Pi-\cot ^{-1} 3\right)+\frac{1}{2}\left[\log \left[1+x^{2}\right]\right]_{-3}{ }^{3} \)
\( =3 \cot ^{-1} 3+3 \pi-3 \cot ^{-1} 3+\frac{1}{2}[\log 10-\log 10] \)
\( =3 \pi \)