Question

$\int \sqrt{\frac{\cos x-{\cos }^{3}x}{1-{\cos }^{3}x}}dx=$

• A. $\frac{2}{3}{\sin }^{-1}\left({\cos }^{\frac{3}{2}}x\right)+c$
• B. $\frac{3}{2}{\sin }^{-1}\left({\cos }^{\frac{3}{2}}x\right)+c$
• C. $\frac{2}{3}{\cos }^{-1}\left({\cos }^{\frac{3}{2}}x\right)+c$
• D. $\frac{3}{2}{\cos }^{-1}\left({\cos }^{\frac{3}{2}}x\right)+c$

Answer 1

Answer: (C)

$\frac{2}{3}{\cos }^{-1}\left({\cos }^{\frac{3}{2}}x\right)+c$

$\int \sqrt{\frac{cosx-co{s}^{3}x}{1-co{s}^{3}x}}dx$

$\int \frac{\sqrt{cosx}sinx}{\sqrt{1-(co{s}^{3}x{)}^2}}dx=\frac{-2}{3}\int \frac{dcos{x}^{\frac{3}{2}}}{\sqrt{1-(co{s}^{\frac{3}{2}}x{)}^2}}$

$=\frac{-2}{3}si{n}^{-1}\left(co{s}^{\frac{3}{2}}x\right)+c$

$=\frac{2}{3}co{s}^{-1}\left(co{s}^{\frac{3}{2}}x\right)-\frac{\pi }{3}+c$

$=\frac{2}{3}co{s}^{-1}\left(co{s}^{\frac{3}{2}}x\right)+c_1$

$\because \frac{\pi }{2}-si{n}^{-1}\left(x\right)=co{s}^{-1}\left(x\right)$