Given that $\int_1^6 \mathrm{f}(x) \, \mathrm{d}x = 14$ and \(\int_1^3 \mathrm{f}(x) \, \math

Question

Given that $\int_1^6 \mathrm{f}(x) \, \mathrm{d}x = 14$ and $\int_1^3 \mathrm{f}(x) \, \mathrm{d}x = 8$, find $\int_3^6 \mathrm{f}(x) \, \mathrm{d}x$ ,[1] $\int_1^6 (4x - 3\mathrm{f}(x)) \, \mathrm{d}x$ .[2] It is given that $\mathrm{f}(x) = \ln(\sin 2x)$. Show that $\mathrm{f}''(x) + [\mathrm{f}'(x)]^2 + 4 = 0$.[5]

Accepted Answer

(i) $x^6(7\ln 7x+1)$ (ii) $\frac{x^7\ln 7x}{7}-\frac{x^7}{49}+C$ (iii) $\frac{x^k\ln kx}{k}-\frac{x^k}{k^2}+C$

2025 SST PRELIMS P1 Q8

2025 SST PRELIMS P1 Q8

Solution:

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