Edexcel A-Level Mathematics A2 Fp3 4 3 Integration Questions

Practise standard integrals from completed squares and trig or hyperbolic substitutions, with exact antiderivatives and bounds.

Exam points

[Maximum number: 6]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use the substitution $x=3\sec\theta$ to determine the exact value of

\int_{2\sqrt3}^{6}\frac{18}{(x^2-9)^{3/2}}\,dx

Give your answer in the form $A+B\sqrt3$ , where A and B are constants to be found.

x=6\Rightarrow \theta=\frac\pi3,\qquad x=2\sqrt3\Rightarrow \theta=\frac\pi6

Since $dx=3\sec\theta\tan\theta\,d\theta$ ,

\int\frac{18}{(x^2-9)^{3/2}}\,dx=\int\frac{54\sec\theta\tan\theta}{(9\sec^2\theta-9)^{3/2}}\,d\theta=2\int\frac{\cos\theta}{\sin^2\theta}\,d\theta

2\int\cosec\theta\cot\theta\,d\theta=-2\cosec\theta

[-2\cosec\theta]_{\pi/6}^{\pi/3}=4-\frac43\sqrt3

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