Find $$\(\int \sin x\left(\operatorname{cosec} \frac{1}{2} x-\sec \frac{1}{2} x\right) \mathrm{d} x\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s20_qp_21 Year：2020 Question No：6(c)
Answer:
Express integrand as \(\sqrt{8} \cos \left(\frac{1}{2} x+\frac{1}{4} \pi\right)\) or as \(2 \cos \frac{1}{2} x-2 \sin \frac{1}{2} x\)
Integrate to obtain \(k \sin \left(\frac{1}{2} x+\frac{1}{4} \pi\right)\) or \(k_{1} \sin \frac{1}{2} x+k_{2} \cos \frac{1}{2} x\)
Obtain correct \(2 \sqrt{8} \sin \left(\frac{1}{2} x+\frac{1}{4} \pi\right)\) or \(4 \sin \frac{1}{2} x+4 \cos \frac{1}{2} x\)
Integrate to obtain \(k \sin \left(\frac{1}{2} x+\frac{1}{4} \pi\right)\) or \(k_{1} \sin \frac{1}{2} x+k_{2} \cos \frac{1}{2} x\)
Obtain correct \(2 \sqrt{8} \sin \left(\frac{1}{2} x+\frac{1}{4} \pi\right)\) or \(4 \sin \frac{1}{2} x+4 \cos \frac{1}{2} x\)
Knowledge points:
2.5.1 extend the idea of 'reverse differentiation' to include the integration of (Knowledge of the general method of integration by substitution is not required.)
2.5.2 use trigonometrical relationships in carrying out integration
Solution:
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