Hence find the exact value of $$\(\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \frac{1-\cos 2 \theta}{1+\cos 2 \theta} \mathrm{d} \theta\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_31 Year：2021 Question No：4(b)
Answer:
Express \(\tan ^{2} \theta\) in terms of \(\sec ^{2} \theta\)
Integrate and obtain terms \(\tan \theta-\theta\)
Substitute limits correctly in an integral of the form \(a \tan \theta+b \theta\), where \(a b \neq 0\)
Obtain answer \(\frac{2}{3} \sqrt{3}-\frac{1}{6} \pi\)
Integrate and obtain terms \(\tan \theta-\theta\)
Substitute limits correctly in an integral of the form \(a \tan \theta+b \theta\), where \(a b \neq 0\)
Obtain answer \(\frac{2}{3} \sqrt{3}-\frac{1}{6} \pi\)
Knowledge points:
3.5.1 extend the idea of ‘reverse differentiation’ to include the integration of (Including examples such as.)
3.5.2 use trigonometrical relationships in carrying out integration
Solution:
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