The equation of a curve is $$\(y=x^{-\frac{2}{3}} \ln x\)$$ for $$\(x>0\)$$. The curve has one stationary point. Show that $$\(\int_{1}^{8} y \mathrm{~d} x=18 \ln 2-9\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_31 Year：2021 Question No：9(b)
Answer:
Commence integration and reach \(a x^{\frac{1}{3}} \ln x+b \int x^{\frac{1}{3}} \cdot \frac{1}{x} \mathrm{~d} x\)
Obtain \(3 x^{\frac{1}{3}} \ln x-3 \int x^{\frac{1}{3}} \cdot \frac{1}{x} \mathrm{~d} x\)
Complete the integration and obtain \(3 x^{\frac{1}{3}} \ln x-9 x^{\frac{1}{3}}\)
Use limits correctly in an expression of the form \(p x^{\frac{1}{3}} \ln x+q x^{\frac{1}{3}}(p q \neq 0)\)
Obtain \(18 \ln 2-9\) from full and correct working
Obtain \(3 x^{\frac{1}{3}} \ln x-3 \int x^{\frac{1}{3}} \cdot \frac{1}{x} \mathrm{~d} x\)
Complete the integration and obtain \(3 x^{\frac{1}{3}} \ln x-9 x^{\frac{1}{3}}\)
Use limits correctly in an expression of the form \(p x^{\frac{1}{3}} \ln x+q x^{\frac{1}{3}}(p q \neq 0)\)
Obtain \(18 \ln 2-9\) from full and correct working
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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