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2025 YIJC P1 Q8

2025 YIJC P1 Q8

Junior College 2
8 marks
  1. Find the exact value of \(\displaystyle \displaystyle\int_0^{\frac{1}{4}\pi} \mathrm{e}^x \cos 2x \, \mathrm{d}x\).[4]
    1. Show that \(\frac{\mathrm{d}}{\mathrm{d}x} \ln(\tan x) = \frac{k}{\sin 2x}\), where \(k\) is a constant to be determined.[2]
    2. Hence find \(\displaystyle \displaystyle\int \mathrm{cosec} 2x [\ln(\tan x) + \cos 2x] \, \mathrm{d}x\).[2]

Solution:

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Answer:

(a) \(\frac{2e^{\frac\pi4}-1}{5}\).

(b)(i) \(k=2\).

(b)(ii) \(\frac14[\ln(\tan x)]^2+\frac12\ln|\sin2x|+C\), on intervals where \(\tan x>0\).

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