<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">State the derivative of \(\mathrm{e}^{\tan x}\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Hence, find \(\int \mathrm{e}^{\tan x} (\sec^3 x)(\sin x) \,\mathrm{d}x\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Write down constants \(A\) and \(B\) such that, for all values of \(x\), \(1 - 4x = A(3 - 8x) + B\). Hence, find </div><div class="list-item-paragraph"><p>\(\int \frac{1 - 4x}{\sqrt{1 - 4x^2 + 3x}} \,\mathrm{d}x.\)</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use the substitution \(u = \sqrt{1 + x}\) to find the exact value of \(\int_{1}^{3} \frac{x^2}{\left(\sqrt{1 + x}\right)^3} \,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Write down constants \(A\) and \(B\) such that, for all values of \(x\), \(1 - 4x = A(3 - 8x) + B\). Hence, find 

<p>\(\int \frac{1 - 4x}{\sqrt{1 - 4x^2 + 3x}} \,\mathrm{d}x.\)</p>

Use the substitution \(u = \sqrt{1 + x}\) to find the exact value of \(\int_{1}^{3} \frac{x^2}{\left(\sqrt{1 + x}\right)^3} \,\mathrm{d}x\).

2024 VJC Promo Q10

2024 VJC Promo Q10

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