2025 ACJC P2 Q5

By using the substitution \({x = a\tan\theta}\), show that

\({\displaystyle\displaystyle \displaystyle\int \frac{1}{x\sqrt{x^2 + a^2}} \mathrm{d}x = \frac{1}{a}\ln\frac{x}{\sqrt{x^2 + a^2} + a} + c,}\)

where \({x > 0}\) and \({a}\) is a positive constant.[4]

At time \({t}\) in a chemical reaction, \({x}\) kg of substance \(X\) and \({y}\) kg of substance \(Y\) are present. Initially, there is \(4\) kg of \(X\) and \(5\) kg of \(Y\).

The variables \({x}\) and \({y}\) satisfy the equations \({\frac{\mathrm{d}x}{\mathrm{d}t} = xy}\) and \({\frac{\mathrm{d}y}{\mathrm{d}t} = x^2}\).