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Partial fractions antiderivative fixed by one given value: find its value at another point

This type of question, asked 2 times: 25 Jan 2023, 2 Apr 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    25 Jan 2023

    71
    Let f(x)=∫2x(x2+1)(x2+3)dxf(x) = \int\frac{2x}{\left( x^{2}+ 1 \right)\left( x^{2}+ 3 \right)}dx. If f(3)=12(log⁡e5−log⁡e6)f(3) =\frac{1}{2}\left( \log_{e}5 -\log_{e}6 \right), then f(4)f(4) is equal to

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  2. Question 2

    2 Apr 2026 S2

    69
    Let f(x)=∫(16x+24x2+2x−15)dxf(x) = \int\left( \frac{16x+ 24}{x^{2}+ 2x- 15} \right)dx. If f(4)=14log⁡e(3)f(4) = 14\log_{e}(3) and f(7)=log⁡e(2α⋅3β),α,β∈Nf(7) =\log_{e}\left( 2^{\alpha}\cdot3^{\beta} \right),\alpha,\beta\in N, then α+β\alpha+\beta is equal to:

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