Step 1: Understanding the Concept:
The absolute value function \( |\sin x| \) is always non-negative. We need to split the integral where the sign of \( \sin x \) changes, or use the property of even functions.
Step 2: Detailed Explanation:
The function \( f(x) = |\sin x| \) is an even function because \( f(-x) = |\sin(-x)| = |-\sin x| = |\sin x| = f(x) \).
For even functions:
\[ \int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx \]
So, \( \int_{-\pi/2}^{\pi/2} |\sin x| dx = 2 \int_{0}^{\pi/2} |\sin x| dx \).
In the interval \( [0, \pi/2] \), \( \sin x \) is non-negative, so \( |\sin x| = \sin x \).
\[ = 2 \int_{0}^{\pi/2} \sin x dx \]
\[ = 2 [-\cos x]_{0}^{\pi/2} \]
\[ = 2 [-\cos(\pi/2) - (-\cos(0))] \]
\[ = 2 [0 + 1] = 2 \].
Step 3: Final Answer:
The value of the integral is 2.