If $\int \sqrt{\cos ecx+1}dx=kfog\left(x\right)+c,wherek$ is a real constant, then $k=-2,f\left(x\right)={\cot }^{-1}x,g\left(x\right)=\sqrt{\cos ecx-1}$$k=-2,f\left(x\right)={\tan }^{-1}x,g\left(x\right)=\sqrt{\cos ecx-1}$$k=2,f\left(x\right)={\tan }^{-1}xg\left(x\right)=\frac{\cot x}{\sqrt{\cos ecx-1}}$$k=2,f\left(x\right)={\cot }^{-1}xg\left(x\right)=\frac{\cot x}{\sqrt{\cos ecx-1}}$

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