The diagram shows the curve $$\(y=\frac{\ln x}{x^{4}}\)$$ and its maximum point $$\(M\)$$. By using integration by parts, show that for all $$\(a>1, \int_{1}^{a} \frac{\ln x}{x^{4}} \mathrm{~d} x<\frac{1}{9}\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_33 Year：2021 Question No：8(b)
Answer:
Commence integration and reach \(a x^{-3} \ln x+b \int x^{-3} \cdot \frac{1}{x} \mathrm{~d} x\)
Obtain \(-\frac{1}{3} x^{-3} \ln x+\frac{1}{3} \int x^{-3} \cdot \frac{1}{x} \mathrm{~d} x\)
Complete integration and obtain \(-\frac{1}{3} x^{-3} \ln x-\frac{1}{9} x^{-3}\)
Substitute limits correctly, having integrated twice
Obtain answer \(\frac{1}{9}-\frac{1}{3} a^{-3} \ln a-\frac{1}{9} a^{-3}\)
Justify the given statement
Obtain \(-\frac{1}{3} x^{-3} \ln x+\frac{1}{3} \int x^{-3} \cdot \frac{1}{x} \mathrm{~d} x\)
Complete integration and obtain \(-\frac{1}{3} x^{-3} \ln x-\frac{1}{9} x^{-3}\)
Substitute limits correctly, having integrated twice
Obtain answer \(\frac{1}{9}-\frac{1}{3} a^{-3} \ln a-\frac{1}{9} a^{-3}\)
Justify the given statement
Knowledge points:
3.2.1 understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
3.5.5 recognise when an integrand can usefully be regarded as a product, and use integration by parts
Solution:
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