The constant $$\(a\)$$ is such that $$\(\int_{1}^{a} \frac{\ln x}{\sqrt{x}} \mathrm{~d} x=6\)$$. Show that $$\(a=\exp \left(\frac{1}{\sqrt{a}}+2\right)\)$$. [ $$\(\exp (x)\)$$ is an alternative notation for $$\(\mathrm{e}^{x}\)$$.] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w21_qp_31 Year：2021 Question No：8(a)
Answer:
Commence integration and reach \(a \sqrt{x} \ln x+b \int \sqrt{x} \cdot \frac{1}{x} \mathrm{~d} x\), or equivalent
Obtain \(2 \sqrt{x} \ln x-\int 2 \sqrt{x} \cdot \frac{1}{x} \mathrm{~d} x\), or equivalent
Obtain integral \(2 \sqrt{x} \ln x-4 \sqrt{x}\), or equivalent
Substitute limits and equate result to 6
Rearrange and obtain \(a=\exp \left(\frac{1}{\sqrt{a}}+2\right)\)
Obtain \(2 \sqrt{x} \ln x-\int 2 \sqrt{x} \cdot \frac{1}{x} \mathrm{~d} x\), or equivalent
Obtain integral \(2 \sqrt{x} \ln x-4 \sqrt{x}\), or equivalent
Substitute limits and equate result to 6
Rearrange and obtain \(a=\exp \left(\frac{1}{\sqrt{a}}+2\right)\)
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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