Value of Definite Integral Since $\sin\left(\frac{1}{x}\right)$ is bounded and continuous on $(0, 1]$ the proper (Riemann-)integral $\int_0^1\sin\left(\frac{1}{x}\right)dx$ exists and is independent of the value prescribed at $x=0$. What is the value of this integral?

There is no solution in terms of elementary functions. Mathematica gives : 
$$\int_0^1\sin({1\over x})\,dx=\sin(1)−\hbox{CosIntegral}(1),$$ where CosIntegral($x$) is defined as $−\int_x^{\infty}{\cos t\over t}\,dt$
