Question

Evaluate the following integral:

\[\int\limits_{- \pi/2}^{\pi/2} \left\{ \sin \left| x \right| + \cos \left| x \right| \right\} dx\]

 

Answer 1

\[\int_\frac{- \pi}{2}^\frac{\pi}{2} \left\{ \sin \left| x \right| + \cos \left| x \right| \right\} d x\]
\[Since\, f\left( - x \right) = \sin \left| - x \right| + \cos \left| - x \right| = \sin \left| x \right| + \cos \left| x \right| = f\left( x \right)\]
\[So, f\left( x \right) \text{is an even function} . \]
\[ \therefore I = 2 \int_0^\frac{\pi}{2} \left( \sin x + \cos x \right) dx\]
\[ \Rightarrow I = 2 \left[ - \cos x + \sin x \right]_0^\frac{\pi}{2} \]
\[ \Rightarrow I = 2\left( 0 + 1 + 1 - 0 \right)\]
\[ \Rightarrow I = 4\]