Question

Evaluate the following integral:

\[\int\limits_{- \pi/4}^{\pi/4} \left| \sin x \right| dx\]

Answer 1

\[\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \left| \sin x \right| d x\]
\[\text{We know that}, \left| \sin x \right| = \begin{cases} - \sin x &,& - \frac{\pi}{4} \leq x \leq 0\\\sin x&,& 0 < x \leq \frac{\pi}{4}\end{cases}\]
\[ \therefore I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \left| \sin x \right| d x\]
\[ \Rightarrow I = \int_{- \frac{\pi}{4}}^0 - \sin x dx + \int_0^\frac{\pi}{4} \sin x dx\]
\[ \Rightarrow I = \left[ \cos x \right]_\frac{- \pi}{4}^0 - \left[ \cos x \right]_0^\frac{- \pi}{4} \]
\[ \Rightarrow I = 1 - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} + 1\]
\[ \Rightarrow I = 2 - \frac{2}{\sqrt{2}}\]
\[ \Rightarrow I = 2 - \sqrt{2}\]