Asymptotics of the integral $\int_0^\pi x^n \sin(x)dx $ I am going through de bruijn's book on asymptotic methods. In the end of a chapter on Laplace's method for integrals, there is an exercise to show the following asymptotic: $$\int_0^\pi x^n\sin(x)dx\sim \frac{\pi^{n+2}}{n^2}, n\to\infty $$ I couldn't relate this to the examples in the chapter, where he dealt mainly with integrals of the form $\int_I e^{-tx^2}f(x)dx $, where $t>0$ a real number, and where that width of the interval contribuiting the most for the result was small (here it is of constant length to my understanding, $[1,\pi]$). However I manged to show (using the obviouse bound $\sin(x)<x$) a weaker result. I tried to read through the chapter again, but I have no idea how to do better here. I would very appreciate any hints or sketches of solution.

For any $x\in(0,\pi)$ we have

$$ \frac{\sin x}{x(\pi -x)} = \frac{1}{\pi}+K x(\pi-x),\qquad K\in\left[\frac{1}{\pi^3},\frac{4(4-\pi)}{\pi^4}\right] \tag{1}$$ hence

$$ \int_{0}^{\pi}x^n\sin(x)\,dx = \frac{\pi^{n+2}}{(n+2)(n+3)}+\pi^n O\left(\frac{1}{n^3}\right)\tag{2}$$ due to $\int_{0}^{\pi}x^\alpha(\pi-x)^{\beta}\,dx = \pi^{\alpha+\beta+1}\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}$.