Question

INTEGRATE : (1/x)(sin((1/x)-x)) with limits 1/3 to 3

Answer 1

$$\int_{1/a}^{a} \frac{1}{x} \sin(x - 1/x) dx = - \int_{a}^{1/a} \frac{1}{x} \sin(x - 1/x) dx = |y = 1/x| = - \int_{a}^{1/a} y \sin(1/y - y) d\left(\frac{1}{y}\right) =$$ $$= \int_{1/a}^{a} y \sin(y - 1/y) \frac{1}{y^2} dy = - \int_{1/a}^{a} \frac{1}{y} \sin(y - 1/y) dy = 0$$