A curve has equation $$\(y=\frac{1}{k} x^{\frac{1}{2}}+x^{-\frac{1}{2}}+\frac{1}{k^{2}}\)$$ where $$\(x>0\)$$ and $$\(k\)$$ is a positive constant. It is given instead that $$\(\int_{\frac{1}{4} k^{2}}^{k^{2}}\left(\frac{1}{k} x^{\frac{1}{2}}+x^{-\frac{1}{2}}+\frac{1}{k^{2}}\right) \mathrm{d} x=\frac{13}{12}\)$$. Find the value of $$\(k\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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