<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">State the derivative of \(\mathrm{e}^{\tan x}\).<span class="inline-marks">[1]</span></div></li><li><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><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">\(\int \frac{1 - 4x}{\sqrt{1 - 4x^2} + 3x} \,\mathrm{d}x.\)<span class="inline-marks">[5]</span></div></li><li><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 

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

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