2023TrialShort answer

Q12 (c) (iii)

2 marks

Question

Let

I_n=\int_{e^{-1}}^1(1+\log_e x)^n\,dx

and

J_n=\int_{e^{-1}}^1(\log_e x)(1+\log_e x)^n\,dx

for $n=0,1,2,3,\ldots$ .

Hence find the value of $J_3$ in simplest exact form.

Try the question before revealing the answer.

Solution

\begin{aligned} J_3&=1-(3+2)I_3\\ &=1-5(1-3I_2)\\ &=1-5+15(1-2I_1)\\ &=-4+15-30(1-I_0)\\ &=11-30+30\int_{e^{-1}}^1 1\,dx\\ &=-19+30[x]_{e^{-1}}^1\\ &=-19+30(1-e^{-1})\\ &=11-30e^{-1} \end{aligned}

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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