$$$ \operatorname{Let} \mathrm{f}(x)=\frac{15-6 x}{(1+2 x)(4-x)} $$$. Hence find $$$\int_{1}^{2} \mathrm{f}(x) \mathrm{d} x$$$, giving your answer in the form $$$\ln \left(\frac{a}{b}\right)$$$, where $$$a$$$ and $$$b$$$ are integers. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s21_qp_33 Year：2021 Question No：4(b)

Answer:

Integrate and obtain terms $2 \ln (1+2 x)+\ln (4-x)$
Substitute limits correctly in an integral of the form $a \ln (1+2 x)+b \ln (4-x)$, where $a b \neq 0$
Obtain final answer $\ln \left(\frac{50}{27}\right)$

Knowledge points:

3.2.1 understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)

3.5.1 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.)

Answer:

Knowledge points:

Solution:

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