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 (e) and how it can be improved.[2]