Question

If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:

\[\int\limits_0^{\pi/4} \sin \left\{ x \right\} dx\]

 

Answer 1

\[\text{We have}, \]
\[I = \int\limits_0^{\pi/4} \sin \left\{ x \right\} dx\]
\[\text{We know that}, \]
\[\left\{ x \right\} = x\text{, when }0 < x < \frac{\pi}{4} ..............\left(\text{As }\pi = 3 . 14 \Rightarrow \frac{\pi}{4} = 0 . 785 < 1 \right)\]
\[ \therefore I = \int\limits_0^{\pi/4} \sin x\ dx\]
\[ = \left[ - \cos x \right]_0^\frac{\pi}{4} \]
\[ = - \left( \cos \frac{\pi}{4} - \cos 0 \right)\]
\[ = \cos 0 - \cos \frac{\pi}{4}\]
\[ = 1 - \frac{1}{\sqrt{2}}\]
\[ = \frac{\sqrt{2} - 1}{\sqrt{2}}\]