Integration of $\sin(\frac{1}{x})$

Question

How to find the value of the integral?

$$\begin{align} (1)&&\int_{0}^{1}\sin\left(\frac{1}{x}\right)dx\\\\ (2)&&\int_{0}^{1}\cos\left(\frac{1}{x}\right)dx \end{align}$$

Answer 1

The antiderivatives involve the sine of cosine integral functions. From their definition, the antiderivatives are respectively $$x \sin \left(\frac{1}{x}\right)-\text{Ci}\left(\frac{1}{x}\right)$$ and $$\text{Si}\left(\frac{1}{x}\right)+x \cos \left(\frac{1}{x}\right)$$ So, the integrals are $$\sin (1)-\text{Ci}(1)$$ and $$\text{Si}(1)-\frac{\pi }{2}+\cos (1)$$ and their numerical values are respectively $0.504067$ and $-0.084411$.