Let $$\(\mathrm{f}(x)=\frac{5 a}{(2 x-a)(3 a-x)}\)$$, where $$\(a\)$$ is a positive constant. Hence show that $$\(\int_{a}^{2 a} \mathrm{f}(x) \mathrm{d} x=\ln 6\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics
IGCSE&ALevel
CAIE
Exam No:9709_m21_qp_32 Year:2021 Question No:6(b)

Answer:

Integrate and obtain terms \(\ln (2 x-a)-\ln (3 a-x)\)
Substitute limits correctly in a solution containing terms of the form \(b \ln (2 x-a)\) and \(c \ln (3 a-x)\), where \(b c \neq 0\)
Obtain the given answer showing 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)
3.5.1 extend the idea of ‘reverse differentiation’ to include the integration of (Including examples such as.)
3.5.3 integrate rational functions by means of decomposition into partial fractions (Restricted to types of partial fractions as specified in topic 3.1 above.)

Solution:

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