Specialist Maths — Question 15

QCAA 2021, Paper 1 · 4 marks

Q15 · 2021 · Technology-freeSimple familiar4 marks

QUESTION 15 (4 marks)

Use partial fractions to determine

\int\frac{4x-17}{x^2-x-6}\,dx,

where

x\in\mathbb R

x\ne-2

x\ne3

. Express your answer in the form

\ln|f(x)|+c

WORKED SOLUTION

4 marks

ANSWER

\ln\left|\dfrac{(x+2)^5}{x-3}\right|+c

Worked solution

Since

x^2-x-6=(x-3)(x+2)

\frac{4x-17}{(x-3)(x+2)}=-\frac1{x-3}+\frac5{x+2}.

Therefore

\int\frac{4x-17}{x^2-x-6}dx=-\ln|x-3|+5\ln|x+2|+c=\ln\left|\frac{(x+2)^5}{x-3}\right|+c.

correctly factorises the denominator to establish the form of the partial fraction decomposition

[1 mark]

determines values of A and B

[1 mark]

determines indefinite integral of the fraction

[1 mark]

determines expression in the form

\ln|f(x)|

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

How many marks did you earn?

Compare your working with the guide above.