5 marksIntegration II
CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2 · Question 2(a)(i)
Determine \int \sin x \cos 2x \, dx.
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- 2(a)(ii)Hence, calculate
\int_0^{\frac{\pi}{2}} \sin x \cos 2x \, dx.[2 marks] - 2(b)Let
f(x) = x|x| = \begin{cases} x^2 &; x \ge 0 \\ -x^2 &; x < 0 \end{cases}. Use the trapezium rule with four intervals to calculate the area betweenf(x)…[5 marks] - 2(c)(i)Show that
\frac{2x^2 + 4}{(x^2 + 4)^2} = \frac{2}{x^2 + 4} - \frac{4}{(x^2 + 4)^2}.[6 marks] - 2(c)(ii)Hence, find
\int \frac{2x^2 + 4}{(x^2 + 4)^2} \, dx. Use the substitutionx = 2\tan\theta.[7 marks]
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