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\includegraphics[max width=\textwidth, alt={}, center]{7c125770-1ded-4763-8453-b07ef43e83e9-2_446_601_1969_772} The diagram shows the curve with equation $$x ^ { 3 } + x y ^ { 2 } + a y ^ { 2 } - 3 a x ^ { 2 } = 0$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\).

1. By differentiating \(\frac { 1 } { \cos x }\), show that the derivative of \(\sec x\) is \(\sec x \tan x\). Hence show that if \(y = \ln ( \sec x + \tan x )\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x\).
2. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), find the exact value of $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { \left( 3 + x ^ { 2 } \right) } } \mathrm { d } x$$ expressing your answer as a single logarithm.