Question

If $\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x=k \cos ^{-1}\left(t^{3 / 2}\right)+c$ and $0

• A. $(1 / 3, \cos x )$
• B. $(2 / 3, \sin x)$
• C. $(1 / 3, \sin x)$
• D. $(2 / 3, \cos x)$

Answer 1

Answer: (D)

$\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x=\int \sqrt{\frac{\cos x \cdot \sin ^{2} x}{1-\cos ^{3} x}} d x$
$=\int \frac{\sqrt{\cos x}}{\sqrt{1-\left(\cos ^{3 / 2} x\right)^{2}}} \sin x \cdot d x$.
Put $\cos ^{3 / 2} x=y$.
Then $-\frac{3}{2} \cos ^{1 / 2} x \cdot \sin x \cdot d x=d y$.
$\therefore$ above integral
$=\int \frac{-1}{\sqrt{1-t^{2}}} \frac{2}{3} d y=\frac{2}{3} \cdot \cos ^{-1} y$.