<p><strong>(a) </strong>Find \(\int \cos (3 \ln x) \, \mathrm{d}x\).<span class="inline-marks">[4]</span></p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Let \(I\) be the indefinite integral \(\int \frac{\mathrm{P}(x)}{1-\sqrt{x}} \, \mathrm{d}x\), \(0 &lt; x &lt; 1\), where \(\mathrm{P}(x)\) is a polynomial in \(x\).</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find \(I\) when \(\mathrm{P}(x) = 1 - x\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By using the substitution \(u = 1 - \sqrt{x}\), find \(I\) when \(\mathrm{P}(x) = 1\).</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>Hence find \(I\) when \(\mathrm{P}(x) = x\).<span class="inline-marks">[2]</span></p>

**(a) **Find \(\displaystyle \int \cos (3 \ln x) \, \mathrm{d}x\).

Let \(I\) be the indefinite integral \(\displaystyle \int \frac{\mathrm{P}(x)}{1-\sqrt{x}} \, \mathrm{d}x\), \(0 < x < 1\), where \(\mathrm{P}(x)\) is a polynomial in \(x\).

Find \(I\) when \(\mathrm{P}(x) = 1 - x\).

By using the substitution \(u = 1 - \sqrt{x}\), find \(I\) when \(\mathrm{P}(x) = 1\).

2025 NJC P1 Q9

2025 NJC P1 Q9