Hence find $$$\int_{1}^{6} \frac{9 x^{3}-6 x^{2}-20 x+1}{3 x+2} \mathrm{~d} x$$$, giving the answer in the form $$$a+\ln b$$$ where $$$a$$$ and $$$b$$$ are integers. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s20_qp_21 Year：2020 Question No：7(b)

Answer:

Integrate to obtain at least $k_{1} x^{3}$ and $k_{2} \ln (3 x+2)$ terms
Obtain $x^{3}-2 x^{2}-4 x+3 \ln (3 x+2)$
(FT from quotient in part (a))
Apply limits correctly
Apply appropriate logarithm properties correctly
Obtain $125+\ln 64$

Knowledge points:

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

2.5.1 extend the idea of 'reverse differentiation' to include the integration of (Knowledge of the general method of integration by substitution is not required.)

Answer:

Knowledge points:

Solution:

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