# Integrate: $\int \cos^{-1} (-3x +4x^3)dx$

$(a)\;3(\theta \sin \theta -\sin \theta)+c \\(b)\;3 (\theta \sin \theta-\cos \theta) +c \\(c)\; 3(\theta \cos \theta - \sin \theta )+c \\ (d)\;6(\theta \cos \theta - \sin \theta )+c$

Put $x= \cos \theta, dx = -\sin \theta d \theta$
$\qquad= \int \cos^{-1} (-3 \cos \theta+ 4 \cos^3 \theta) (-\sin \theta ) d \theta$
$\qquad= \int \cos ^{-1} (\cos 3 \theta )(-\sin \theta )d \theta$
$\qquad= -3 \int Q. \sin \theta. d \theta$
$\qquad= -3 \{ (-\cos \theta ). \theta + \int \cos \theta d \theta \}$
$\qquad= -3 \{ (-) \theta \cos \theta + \sin \theta \}+c$
$\qquad= 3 \{ \theta \cos \theta - \sin \theta \}+c$
Hence c is the correct answer.