A curve has equation $$$y=\mathrm{f}(x)$$$ where $$$\mathrm{f}(x)=\frac{4 x^{3}+8 x-4}{2 x-1}$$$. Divide $$$4 x^{3}+8 x-4$$$ by $$$(2 x-1)$$$, and hence find $$$\int \mathrm{f}(x) \mathrm{d} x$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w20_qp_23 Year：2020 Question No：8(b)

Answer:

Carry out division to obtain quotient of form $2 x^{2}+k x+m$
Obtain correct quotient $2 x^{2}+x+\frac{9}{2}$
Obtain remainder $\frac{1}{2}$
Integrate to obtain at least $k_{1} x^{3}$ and $k_{2} \ln (2 x-1)$ terms
Obtain $\frac{2}{3} x^{3}+\frac{1}{2} x^{2}+\frac{9}{2} x+\frac{1}{4} \ln (2 x-1)$ as final answer

Knowledge points:

2.1.2 divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero)

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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