Integration — Question 6
Original QCE Vault practice · 4 marks
Q6 · Practice questionSimple familiar4 marks
QUESTION 6 (4 marks)
Consider the function
f(x)=(x+1)(x+2)3x+5,x>−1.a)Express f(x) in partial fractions.
b)Hence evaluate ∫01f(x)dx, giving your answer in exact form.
WORKED SOLUTION
Practice marking scheme
4 marksANSWER∫01f(x)dx=ln6 Sample response and mark allocation
Let
(x+1)(x+2)3x+5=x+1A+x+2B. Then
3x+5=A(x+2)+B(x+1), so
A+B=3 and
2A+B=5.
Forms equations for the partial-fraction constants.[1 mark]
A=2 and
B=1, hence
f(x)=x+12+x+21.
Determines the partial fractions.[1 mark]
∫f(x)dx=2ln(x+1)+ln(x+2)+c for
x>−1.
Determines an antiderivative.[1 mark]
[2ln(x+1)+ln(x+2)]01=2ln2+ln3−ln2=ln6.
Evaluates the definite integral exactly.[1 mark]
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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