$\begin{aligned} & \int_{-1}^1 \frac{\sin x-x^2}{3-|x|} d x \\ & =\int_{-1}^1 \frac{\sin x}{3-|x|} d x-\int_{-1}^1 \frac{x^2}{3-|x|} \\ & =0-2 \int_0^1 \frac{x^2}{3-|x|} d x\left[\because \frac{\sin x}{3-|x|} \text { is odd and }\right. \\ & \frac{x^2}{3-|x|} \text { is even ] }\end{aligned}$
$\begin{aligned} & =-2 \int_0^1 \frac{x^2 d x}{3-|x|}=2 \int_0^1 \frac{x^2}{x-3} d x \\ & =2 \int_0^1\left(x+3+\frac{9}{x-3}\right) d x \\ & =2\left[\frac{x^2}{2}+3 x+9 \log |x-3|\right]_0^1 \\ & =\left[7+18 \log \frac{2}{3}\right]\end{aligned}$