<p>It is given that \(\mathrm{f}(x) = \frac{1}{4+9x^2}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find \(\int \mathrm{f}(x) \, \mathrm{d}x\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the binomial expansion for \(\mathrm{f}(x)\), up to and including the term in \(x^4\). Give the coefficients as exact fractions in their simplest form.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the Maclaurin series for \(\tan^{-1} \frac{3x}{2}\). Give the coefficients as exact fractions in their simplest form.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Use your series from part (c) to estimate \(\int_{0}^{0.5} \tan^{-1} \frac{3x}{2} \, \mathrm{d}x\), correct to \(3\) decimal places.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Use your calculator to find \(\int_{0}^{0.5} \tan^{-1} \frac{3x}{2} \, \mathrm{d}x\), correct to \(3\) decimal places.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Comparing your answers to parts (d) and (e), comment on the accuracy of your estimate in (e) and how it can be improved.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that \(\mathrm{f}(x) = \frac{1}{4+9x^2}\).

Find \(\displaystyle \int \mathrm{f}(x) \, \mathrm{d}x\).

Find the binomial expansion for \(\mathrm{f}(x)\), up to and including the term in \(x^4\). Give the coefficients as exact fractions in their simplest form.

Hence, find the Maclaurin series for \(\tan^{-1} \frac{3x}{2}\). Give the coefficients as exact fractions in their simplest form.

Use your series from part (c) to estimate \(\displaystyle \int_{0}^{0.5} \tan^{-1} \frac{3x}{2} \, \mathrm{d}x\), correct to \(3\) decimal places.

Use your calculator to find \(\displaystyle \int_{0}^{0.5} \tan^{-1} \frac{3x}{2} \, \mathrm{d}x\), correct to \(3\) decimal places.

Comparing your answers to parts (d) and (e), comment on the accuracy of your estimate in (e) and how it can be improved.

2025 JPJC P1 Q9

2025 JPJC P1 Q9