Using the substitution $x=\frac{{{\cos }^{2}} heta }{k}$, find the exact value of $\displaystyle\in

Question

Using the substitution $x=\frac{{{\cos }^{2}}	heta }{k}$, find the exact value of $\displaystyle\int_{0}^{\frac{1}{2k}}{\sqrt{\frac{kx}{1-kx}}}\,\mathrm{d}x$ in terms of $k$, where $k$ is a positive constant. [5]

Accepted Answer

$\frac1k\left(\frac{\pi}{4}-\frac12\right)$

2021 EJC P1 Q2

2021 EJC P1 Q2

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