Question

Write the value of the integral $\underset{0}{\overset{\mathrm{\pi }/4}{\int }}\sin \left\{x\right\}dx,$ where {} denotes the fractional part function.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ I=\underset{0}{\overset{\mathrm{\pi }/4}{\int }}\sin \left\{x\right\}dx \\ \mathrm{We}\mathrm{know}\mathrm{that}, \\ \left\{x\right\}=x,\mathrm{when}0<x<\frac{\mathrm{\pi }}{4}\left(\mathrm{As}\mathrm{\pi }=3.14\Rightarrow \frac{\mathrm{\pi }}{4}=0.785<1\right) \\ \therefore I=\underset{0}{\overset{\mathrm{\pi }/4}{\int }}\sin xdx \\ ={\left[-\cos x\right]}_0^{\frac{\mathrm{\pi }}{4}} \\ =-\left(\cos \frac{\mathrm{\pi }}{4}-\cos 0\right) \\ =\cos 0-\cos \frac{\mathrm{\pi }}{4} \\ =1-\frac{1}{\sqrt{2}} \\ =\frac{\sqrt{2}-1}{\sqrt{2}} \end{gathered}$$