Find $\int \frac{1}{x}(\ln x)^5 \,\mathrm{d}x$.[2] Use the substitution $u = e^{2x}$ to find \(\

Question

Find $\int \frac{1}{x}(\ln x)^5 \,\mathrm{d}x$.[2] Use the substitution $u = e^{2x}$ to find $\int \frac{e^{2x}}{1 + 4e^{4x}} \,\mathrm{d}x$.[4] Evaluate $\int_0^2 \frac{e^{2x}}{1 + 4e^{4x}} \,\mathrm{d}x$, giving the answer correct to $3$ decimal places.[1]

Accepted Answer

(i) $\frac{(\ln x)^6}{6}+C$ (ii) $\frac14\tan^{-1}(2e^{2x})+C$ (iii) $0.114$ (3 d.p.)

2018 TGM P1 Q4

2018 TGM P1 Q4

Solution:

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