Question

Evaluate the following integral:

\[\int\limits_0^{\pi/2} \left| \cos 2x \right| dx\]

Answer 1

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