2025 JPJC P1 Q9
2025 JPJC P1 Q9
11 marks
It is given that \(\mathrm{f}(x) = \frac{1}{4+9x^2}\).
- Find \(\displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x\).[2]
- 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.[2]
- Hence, find the Maclaurin series for \(\tan^{-1} \frac{3x}{2}\). Give the coefficients as exact fractions in their simplest form.[3]
- Use your series from part (c) to estimate \(\displaystyle \displaystyle\int_{0}^{0.5} \tan^{-1} \frac{3x}{2} \, \mathrm{d}x\), correct to \(3\) decimal places.[1]
- Use your calculator to find \(\displaystyle \displaystyle\int_{0}^{0.5} \tan^{-1} \frac{3x}{2} \, \mathrm{d}x\), correct to \(3\) decimal places.[1]
- Comparing your answers to parts (d) and (e), comment on the accuracy of your estimate in (d) and how it can be improved.[2]
Solution:
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Answer:(a) \(\frac16\tan^{-1}\frac{3x}{2}+C\)
(b) \(\mathrm f(x)=\frac14-\frac9{16}x^2+\frac{81}{64}x^4-\cdots\), \(|x|<\frac23\)
(c) \(\tan^{-1}\frac{3x}{2}=\frac32x-\frac98x^3+\frac{243}{160}x^5-\cdots\)
(d) \(0.174\)
(e) \(0.173\)
(f) Accurate to \(2\), not \(3\), decimal places; include further series terms.
(b) \(\mathrm f(x)=\frac14-\frac9{16}x^2+\frac{81}{64}x^4-\cdots\), \(|x|<\frac23\)
(c) \(\tan^{-1}\frac{3x}{2}=\frac32x-\frac98x^3+\frac{243}{160}x^5-\cdots\)
(d) \(0.174\)
(e) \(0.173\)
(f) Accurate to \(2\), not \(3\), decimal places; include further series terms.
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