Given $$ I_n = \int_0^{n\pi} \frac{\sin x}{1 + x} \, \mathrm{d}x, \qquad n = 1,2,3,4, $$ how can I arrange the four integrals in increasing order of magnitude?
note- I can only say that sine function is positive in $(0,\pi)$, $(2\pi,3\pi)$ and negative in $(\pi,2\pi)$, $(3\pi,4\pi)$ but the function tries to integrate from $[0,n\pi]$ (for the given $n$). So it’s not clear to me what can be said about the sequences $I_{n}$. Thank you!
Since $\sin(x)$ is oscillating and the amplitude $1/(1+x)$ is decreasing, we can easily see $I_n$ has a damped oscillation pattern with period $\pi$. The key is to compare the 'magnitude' of these oscillations on each interval of length $\pi$, here is the idea:
For $m\in\mathbb Z^+$, let
$$A_m=\left|\int_{m\pi}^{(m+1)\pi}\frac{\sin(x)}{1+x}dx\right|.$$
Since $\sin(x+\pi)=-\sin(x)$, it follows
$$\int_{m\pi}^{(m+1)\pi}\frac{\sin(x)}{1+x}dx=\int_{(m-1)\pi}^{m\pi}\frac{\sin(x+\pi)}{1+x+\pi}dx=-\int_{(m-1)\pi}^{m\pi}\frac{\sin(x)}{1+x+\pi}dx$$
and therefore $A_m<A_{m-1}$. Using this and the fact $I_n=\sum_{m=0}^{n-1} (-1)^mA_m$, you can easily conclude that $|I_1|>|I_3|>|I_4|>|I_2|$.