<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the substitution \(t = \mathrm{e}^x\) to show that \(\int \mathrm{e}^{2x} \tan^{-1}(\mathrm{e}^x) \, \mathrm{d}x = \int t \tan^{-1}(t) \, \mathrm{d}t\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the exact value of \(\int_0^1 \mathrm{e}^{2x} \tan^{-1}(\mathrm{e}^x) \, \mathrm{d}x\).</span><span class="inline-marks">[5]</span></div></li></ol>

Use the substitution \(t = \mathrm{e}^x\) to show that \(\displaystyle\displaystyle \int \mathrm{e}^{2x} \tan^{-1}(\mathrm{e}^x) \, \mathrm{d}x = \displaystyle\displaystyle \int t \tan^{-1}(t) \, \mathrm{d}t\).

Hence find the exact value of \(\displaystyle \int_0^1 \mathrm{e}^{2x} \tan^{-1}(\mathrm{e}^x) \, \mathrm{d}x\).

2025 DHS P2 Q2

2025 DHS P2 Q2