Does the integral $\int_0^1\sin(1/x)\, \mathrm{d}x$ converge?

Question

I am interested in proving that the integral $$\int_{0^+}^1\sin\left(\frac{1}{x}\right)\,\mathrm d x$$ converges. Can someone show me step by step please?

Answer 1

Hint. Make the substitution $xy=1.$ Then the integral becomes $$\int_1^{+\infty}\frac{\sin y}{y^2}\mathrm dy,$$ which you should be able to show converges.