Is there a reduction formula for $I_n=\int_{0}^{n\pi}\frac{\sin x}{1+x}\,dx$? I haven't been able to manipulate this integral. I need to find the value of $I_n$ for $n=1,2,3,4$ and arrange them in ascending order.

Since the sine function oscillates with zeros at integer multiples of $\pi$, we introduce
$$
K_j=\int_{(j-1)\pi}^{j\pi}\frac{\sin x}{1+x}\,dx,\quad j\in\{1,2,3,4\}.
$$
Then, since $x\mapsto 1/(1+x)$ is decreasing and since $\sin(x+\pi)=-\sin(x)$ we find that
$$
K_1>-K_2>K_3>-K_4>0.
$$
The integrals we want to compare are given by
$$
I_1=K_1,\quad I_2=K_1+K_2,\quad I_3=K_1+K_2+K_3,\quad I_4=K_1+K_2+K_3+K_4,
$$
I stop here, and suggest you to try to conclude from the facts above that the correct order is

$$I_2<I_4<I_3<I_1$$

As suggested in the comments, it might be good to have a picture:

