CAIE A-Level Mathematics A2 3.5.4 An Integrand of the Form X Questions

Practise recognising and integrating standard trigonometric, logarithmic and rational integrand forms.

Exam points

recognise standard integrand forms and choose suitable substitutions
integrate logarithmic, trigonometric and rational forms
evaluate definite integrals using standard antiderivatives

[Maximum number: 4]

Hence show that $\int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi}(\operatorname{cosec} 2 \theta-\cot 2 \theta) \mathrm{d} \theta=\frac{1}{2} \ln 2$ .

State integral of the form $\mp \ln \cos \theta$ or $\pm \ln \sec \theta$
*M1
$[-\ln \cos \theta]_{\frac{\pi}{4}}^{\frac{\pi}{3}} \mathrm{OE}$
Use correct limits correctly and insert exact values for the trigonometric
ratios
Need to see evidence of the substitution
Obtain a correct expression, e.g. $-\ln \frac{1}{2}+\ln \frac{1}{\sqrt{2}}$
Obtain $\frac{1}{2} \ln 2$ from correct working
AG (must see an intermediate step)

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CAIE A-Level Mathematics A2 3.5.4 An Integrand of the Form X Questions question 1