# Convergence of $\int_1^\infty \! \sin^\alpha(1/x) \, \mathrm{d}x$ for $\alpha > 1$

How do I show that for $\alpha > 1$ the integral $\displaystyle \int_1^\infty \! \sin^\alpha(1/x) \, \mathrm{d}x$ converges?

I am given the hint:

Compare with the integral $\displaystyle \int_1^\infty \! x^{-\alpha} \, \mathrm{d}x$.

What basically your hint means that if $f$ and $g$ are two non-negative functions such that $f \leq g$ and $\int g$ converges then $\int f$ converges. This is basically the comparison test.
• I think you mean that "...$f$ and $g$ are two nonnegative functions (over the domain of interest)..." – cardinal May 8 '11 at 16:07
The integral $$\int _{1}^{\infty} \frac{1}{x^\alpha} \ \textrm{dx}$$ converges, as show @Chandru1. Moreover, $$\lim_{x\to \infty} \left( \frac{\sin\left( \frac{1}{x} \right)}{\frac{1}{x}} \right)^\alpha = 1,$$ therefore your integral and $\int _{1}^{\infty} \frac{1}{x^\alpha} \ \textrm{dx}$ have the same behavior, i.e., your integral converges.