Find $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^7\ln 7x\right)$.[3] Hence find \(\int x^6\ln 7x\,\math

Question

Find $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^7\ln 7x\right)$.[3] Hence find $\int x^6\ln 7x\,\mathrm{d}x$.[4] Without additional working, write what the integral will be for $x^{k-1}\ln kx$, where $k$ is an integer.[1] In part (iii), assume $k$ is a nonzero integer and $kx>0$. In parts (i) and (ii), $x>0$.

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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