<ol class="abc-list"><li><div class="list-item-paragraph">By using the substitution ${x = a\tan\theta}$, show that</div><div class="list-item-paragraph" style="text-align: center">${\int \frac{1}{x\sqrt{x^2 + a^2}} \mathrm{d}x = \frac{1}{a}\ln\frac{x}{\sqrt{x^2 + a^2} + a} + c,}$</div><div class="list-item-paragraph">where ${x &gt; 0}$ and ${a}$ is a positive constant.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">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 $Y$.</div><div class="list-item-paragraph">The variables ${x}$ and ${y}$ satisfy the equations ${\frac{\mathrm{d}x}{\mathrm{d}t} = xy}$ and ${\frac{\mathrm{d}y}{\mathrm{d}t} = x^2}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$, and by solving this differential equation, show that $y = \sqrt{x^2+9}.$<span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Obtain a differential equation in terms of $x$ and $t$ only and hence find an expression of $t$ in terms of $x$.</span><span class="inline-marks">[4]</span></div></li></ol></li></ol>

By using the substitution $x=a \tan \theta$, show that\n\[\displaystyle\displaystyle \int \frac{1}{x \sqrt{x^2+a^2}} \, \mathrm{d}x = \frac{1}{a} \ln \frac{x}{\sqrt{x^2+a^2}+a} + c,\]\nwhere $x > 0$ and $a > 0$.

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

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

where ${x > 0}$ and ${a}$ is a positive constant.

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 $Y$.\n\nThe variables $x$ and $y$ satisfy the equations $\frac{\mathrm{d}x}{\mathrm{d}t} = xy$ and $\frac{\mathrm{d}y}{\mathrm{d}t} = x^2$.

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 ${\frac{\mathrm{d}y}{\mathrm{d}t} = x^2}$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$, and by solving this differential equation, show that\n\[y = \sqrt{x^2+9}.\]

Obtain a differential equation in terms of $x$ and $t$ only and hence find an expression of $t$ in terms of $x$.

2025 ACJC P2 Q5

2025 ACJC P2 Q5

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