Answer : C
$I = \int \left ( \sin x - \cos x \right )\left (\sin x + \cos x \right )^{3}dx$
$I = \int \left ( \sin x - \cos x \right )\left (\sin x + \cos x \right )\left (\sin x + \cos x \right )^{2}dx$
$I = \int \left (\sin ^{2}x - \cos^{2} x \right )\left (\sin x + \cos x \right )^{2}dx$
$I = - \int \left (\cos^{2} x - \sin ^{2}x \right )\left (\sin^{2} x + \cos^{2} x + 2 \sin x \cos x \right )dx$
$I = - \int \left (\cos 2x \right )\left (1 + \sin 2x \right )dx$
$Let \: 1 + \sin 2x = t$
$Then \: \cos 2x \, dx \: = \: \frac{dt}{2}$
$I = -\int t\: \frac{dt}{2}$
$I = \frac{-\:t^{2}}{4} + K$
$I = \frac{-\: \left ( 1\:+\:\sin 2x \right )^{2}}{4} + K$
$I = \frac{-\: \left ( \sin^2x \:+ \:\cos^2x \:+\: 2\sin x \cos x \right )^{2}}{4} + K$
$I = \frac{-\: \left ( \sin x \:+\: \cos x \right)^{4}}{4} + K$