It is given that $$\(\int_{0}^{\frac{1}{2} \pi}\left(3+4 \cos ^{2} \frac{1}{2} x+k \sin 2 x\right) \mathrm{d} x=10\)$$. Find the exact value of the constant $$\(k\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w20_qp_22 Year：2020 Question No：6(b)
Answer:
Express \(4 \cos ^{2} \frac{1}{2} x\) in the form \(p+q \cos x\)
Obtain correct \(2+2 \cos x\)
Integrate to obtain form \(p x+q \sin x+r \cos 2 x\)
Obtain correct \(5 x+2 \sin x-\frac{1}{2} k \cos 2 x\)
Apply limits correctly, equate to 10 and solve for \(k\)
Obtain \(k=8-\frac{5}{2} \pi\)
Obtain correct \(2+2 \cos x\)
Integrate to obtain form \(p x+q \sin x+r \cos 2 x\)
Obtain correct \(5 x+2 \sin x-\frac{1}{2} k \cos 2 x\)
Apply limits correctly, equate to 10 and solve for \(k\)
Obtain \(k=8-\frac{5}{2} \pi\)
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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