<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact value of \(\int_0^{\frac{1}{4}\pi} \mathrm{e}^x \cos 2x \, \mathrm{d}x\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(\frac{\mathrm{d}}{\mathrm{d}x} \ln(\tan x) = \frac{k}{\sin 2x}\), where&nbsp;\(k\) is a constant to be determined.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find \(\int \mathrm{cosec} 2x [\ln(\tan x) + \cos 2x] \, \mathrm{d}x\).</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Find the exact value of \(\displaystyle \int_0^{\frac{1}{4}\pi} \mathrm{e}^x \cos 2x \, \mathrm{d}x\).

Show that \(\frac{\mathrm{d}}{\mathrm{d}x} \ln(\tan x) = \frac{k}{\sin 2x}\), where \(k\) is a constant to be determined.

Hence find \(\displaystyle \int \mathrm{cosec} 2x [\ln(\tan x) + \cos 2x] \, \mathrm{d}x\).

2025 YIJC P1 Q8

2025 YIJC P1 Q8