The value of ∫_-1^1√(|x|-x) dx is equal to :

The correct answer is Option (4) → \( \displaystyle\frac{2\sqrt{2}}{3} \)

Because of the modulus, the integral splits at x = 0.

For \( x\ge 0 \), \( |x|-x=0 \), so the integrand is 0 there.

For \( x\lt 0 \), \( |x|-x=-2x \), so

    \( \displaystyle \int_{-1}^{1}\sqrt{|x|-x}\,dx=\int_{-1}^{0}\sqrt{-2x}\,dx \)

Let \( t=-x \):

    \( \displaystyle \int_{-1}^{0}\sqrt{-2x}\,dx=\sqrt{2}\int_{0}^{1}t^{1/2}\,dt \)

    \( \displaystyle =\sqrt{2}\cdot\frac{2}{3} \)

    \( \displaystyle =\frac{2\sqrt{2}}{3} \)