$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x \sin x}{1+x^{8}} d x$

Step 1 ：Define the function $f(x) = \frac{x \sin x}{1+x^{8}}$ and the limits of integration $a = -\frac{\pi}{4}$ and $b = \frac{\pi}{4}$

Step 2 ：Choose a value for $n$ (the number of subintervals), for example, $n = 1000$

Step 3 ：Calculate the step size $h = \frac{b-a}{n} = \frac{\frac{\pi}{4} - (-\frac{\pi}{4})}{1000} = 0.0015707963267948967$

Step 4 ：Calculate the $x$ values for each subinterval: $x_i = a + ih$ for $i = 0, 1, \dots, n$

Step 5 ：Use the trapezoidal rule to approximate the integral: $\int_{a}^{b} f(x) dx \approx \frac{h}{2} \sum_{i=1}^{n} (f(x_{i-1}) + f(x_i))$

Step 6 ：Calculate the approximate value of the integral: $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x \sin x}{1+x^{8}} d x \approx 0.29270320329010313$

Step 7 ：\(\boxed{0.2927}\)