Question 1

Paper 1, no calculatorMedium12 marks

Let

f(x)=x+8x2+x−6,x∈R,f(x)=\frac{x+8}{x^2+x-6}, \qquad x\in\mathbb R,

where f(x)f(x) is defined only when its denominator is non-zero.

(a)

Write x2+x−6x^2+x-6 as a product of linear factors, and state the values of xx for which f(x)f(x) is undefined.

[2]
(b)

Write f(x)f(x) in partial fractions.

[4]
(c)

For parts (i) and (ii), consider −3<x<2-3<x<2

(i)

Hence determine an antiderivative FF of ff

[3]
(ii)

Hence evaluate ∫01f(x) dx\displaystyle\int_0^1 f(x)\,dx giving your answer exactly. Explain why the sign of your answer is consistent with the sign of f(x)f(x) on [0,1][0,1]

[3]

More AA 1.5 Partial fractions and linear systems questions