If $\int \sqrt{\frac{\cos x-\cos ^3 x}{\left(1-\cos ^3 x\right)}} d x=f(x)+c$, then $f(x)$ is equal

Question

If $\int \sqrt{\frac{\cos x-\cos ^3 x}{\left(1-\cos ^3 x\right)}} d x=f(x)+c$, then $f(x)$ is equal to:

Professor PC Thomas Classes · Answer

Let $I=\int\sqrt{\frac{cos~x-cos^{3}x}{(1-cos^{3}x)}}dx=\int\frac{\sqrt{cos~x}(\sqrt{1-cos^{3}x})}{\sqrt{1-(cos^{3}x)}}dx$ $\hat{f}cos^{\frac{3}{2}}x=p,then(-\frac{3}{2}cos^{\frac{1}{2}}x~sin~x)dx=dp$ $I=-\frac{2}{3}\int\frac{dp}{\sqrt{1-p^{2}}},=-\frac{2}{3}sin^{-1}(cos^{\frac{3}{2}}x)+C=\frac{2}{3}cos^{-1}(cos^{\frac{3}{2}}x)+C_{1}$