Mathematical Methods Unit 4 Integration Revision
1 Given that f (x) = 5x4 – 4x and f(1) = 2, then f(x) is equal
to:
A x5 – 4x + 2
B x5 – 2x2 + 1
C x5 – 4x2 + 5
d
5 Given that dx (xe3x) = e3x + 3x e3x , an antiderivative of
xe3x is:
1
1
A 3 e3x + 2 x2 e3x
D x5 – 4x2 – 1
1
B 6 x2 e3x
E x5 – 2x2 + 3
C 3e3x + 3x e3x
1
1
D  e3x + x e3x
3
3
1
1
E 9 e3x + 3 x e3x
2 The total area of the regions enclosed by the curve
6 Given that for f(x) = x tan x , f ′ (x) = tan x + x sec2 x,
y = sin 2x and the x-axis between x = 0 and x =  is:
an antiderivative of x sec2 x is:
A 0
A x tan x
B 1
B x tan x + loge (cos x)
C 2
C x tan x + loge (sec x)
D
3
E
1
2
D 2 sec2 x
E x tan x  sec2 x
3 The area of the region enclosed by the curve
y = e2x + 2, the line x = 2 and the x-axis is:.
A 1 + loge 2
1
B 2 e4 – 4
C
a
0
The value of a is:
A 4
B 3
11
C 2
681
e
500
1
D loge 2 + 2e4 + 3
E loge 2 
7 
 2x + 4 dx = , where a is a positive real number.
D 6
E 2
1
5
2e4
3
4 f (x) = 8 sin 4x and f( 4 ) = 1, f(x) =
8 If the curve with equation y = f(x) for which
f (x) = 6x + k has a stationary point at (1, 2), f(x) equals
A 32 cos 4x – 31
A 3x2 – 6x
B 2 cos 4x + 1
B 3x2 + x
C 2 sin 4x + 1
C 3x2 – 6x + 5
D 2 cos 4x + 3
D 3x2 + 6x – 4
E 2 sin 4x + 3
E 3x2 + 6x
Mathematical Methods Integration Revision
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y
9 The graph with equation y = k (x2 – 1) is shown.
The area of the shaded region is equal to 20. The value
of k is equal to:
A –30
B –15
0
2
C 3
1
x
y = k(x2 – 1)
D 15
E 30
10 The graph with equation y = x2 for x > 0 is shown.
When the area between the graph and the x-axis for 0 ≤
x ≤ 3 is approximated by using the upper rectangles and
the partitioning shown, the area is:
A 6
B 9
1
C 92
D 10
E 14
11 The graph with equation y = f(x) is shown.
The total area of the shaded region is equal to:
A

B


C –

D

3
f(x)dx
0
1
0


f(x)dx –
1
0
3
f(x) dx
1

f(x)dx +
 f(x)dx – 
1
1
3
0
0
E –  f(x)dx +
1
0
3
f(x)dx
f(x)dx

3
0
f(x)dx
 12 The graphs

with equations y = x3 and y = 8 are shown.

The area of the shaded region is equal to:
A 64 –
B

8
0
 C 16 

D

E

8
0
8
0
8

0
x3dx
x3dx – 64
2
0 x3dx
x3dx – 16
8dx –

8
0
x3dx


Mathematical Methods Integration Revision

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13 Given that

a5
x4dx = 5 , then
a
0

0
–a
x4dx is equal to:
15 Given
dy
dy
= ae-x + 2 and that when x = 0,
= 5 and y =
dx
dx
A 0
1, when x = 2, y =
a5
B
 5
3
A e2 + 2

2a5
C 5
3
B e2 + 4
a5
5
3
C e2 + 8
D –
D 3e2 + 4
2a5
E – 5
E 3e2 + 8
14 The approximation to  x2 + 1 dx, by using the
2
0
rectangles OABC and CDEF with vertices O(0, 0), A(0,
1), B(1, 1), C(1, 0) and C(1, 0), D(1, 2), E(2, 2), F(2, 0),
is:
14
A 3
B 3
C 4.67
D 2
E 4
Answers
1 E 2 C 3 D 4 D 5 E 6 B 7 E 8 C 9 A 10 E 11 B 12 C 13 B 14 B 15 C
Short-answer questions (technology-free)
1 Find an antiderivative of each of the following:
(
a cos 3x +
b

6
)
1
sin 4x
3
c e4x + 6x
d 6 sin 3x – 4e–2x+2
e cos (2x + 2) + 4
(
f sin 4x +

4
)
1
g 3x + 4
Mathematical Methods Integration Revision
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2 Find f(x) if f (x) = ex/4 and f(0) = 4
3 The graph of the function f: R  R , f(x) = e2x – 3ex + 2 is shown below:
y
y=2
y = f(x)
0
c
x
The graph of f passes through the origin and the point (c, 0).
a Find the exact value of c.
b Find the exact values of the coordinates of the turning point of the graph of f.
c Write an appropriate definite integral which determines the area of the region bounded by the graph of f and the xaxis and find the exact value of this area.
4 a Differentiate x cos 2x and hence find an antiderivative of x sin 2x.
b Evaluate


2
0
x sin 2 x dx.
5 Calculate the exact area enclosed between the curve y = 3 cos (2x) – 3 and the x-axis between the lines x =  and x = 
6 a Differentiate loge (x3 + 1) with respect to x.
b Hence or otherwise evaluate
1

0
3x 2
dx.
x3  1
7 The curve with equation y = f(x) has a gradient function with rule f (x) = 4x2 + k, where k is a constant, and has a
turning point with coordinates (–1, 6). Find:
a the value of k
b the rule f(x)
8 Given that
a

1
b

5
c

5
0
0
1

1
0
f (x ) dx =
f (x ) dx +

1
5

5
1
f (x ) dx = 6, evaluate each of the following:
f (x ) dx
f (x ) dx
[2 f ( x)  5] dx
Mathematical Methods Integration Revision
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9 Find the area of the regions bounded by the graph of y = cos x, the lines with equations x =


and x = and the x-axis.
6
3
[ ]
x
10 The graph of y = 3 cos 2 is shown for x  0, π .
y
A
0
B
x
a Find the equation of the straight line which passes through points A and B.
b Find the area of the shaded region.
Analysis Question
A country road runs alongside an east-west railway line. The road then slowly veers away from the railway line
before turning back at point A and eventually crossing over the line. The railway line can be modelled by the straight
line with equation y = 4 and the road by the curve with equation y  e2x  5ex  4 , where each unit on the x and y
axes represents one kilometre.
a
b
c
d

Show that the road passes through the origin.

Find the exact coordinates of the second point where the road crosses the x-axis.
Find the exact coordinates of the point where the road crosses the railway line.
Use calculus to find the exact coordinates of A and hence state how far A is south of the railway line in
kilometres, correct to 2 decimal places.
A farmer wishes to farm the land bounded by the road and the x-axis.
e
Use calculus to find the exact area of this land.
The farmer approaches the local council with a proposal to purchase this land. The councillors, who are keen to sell
a much larger region, will not allow him to purchase the land that he proposes; instead they offer him all of the land
enclosed by the road, the railway track and the line with equation x = –3 for the bargain price of $50 000 per square
kilometre.
f
i Write down a definite integral whose value will give the cost to the farmer of accepting the councillors’
offer.
ii Hence find the amount, to the nearest $100, that the farmer will need to pay to the council if he accepts
their offer.
[1 + 2 + 2 + 4 + 3 + 3 = 15 marks]
Mathematical Methods Integration Revision
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Extended-response questions
Consider the family of functions fa: [0, )  R, defined by fa(x) = x  a x2/3 where a is a real number, a > 0
y
y = fa(x)
x
a fa (c) = 0 and c  0. Express c in terms of a.
b Determine intervals on which fa is a decreasing function and the intervals on which fa is an increasing function.
c Find the equations to the tangents to the graph of fa at the point where the graph of fa crosses the x-axis. What can be
said about these tangents?
d What is the range of fa?
e Find the exact value of the area of the shaded region.
Mathematical Methods Integration Revision
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Answers to Chapter 12 Test B
Answers to short-answer (technology-free) questions
1
π
1 a 3 sin(3x + 6)
1
b –12 cos 4x
1
c 4 e4x + 3x2
d –2 cos (3x) + 2e–2x + 2
1
e 2 sin (2x + 2) + 4x
1
π
f –4 cos (4x + 4 )
1
g 3 loge (3x + 4)
2 f(x) = 4e x/4
3 a loge 2
3
1
b (loge ( ), – )
2
4
c –

0
e – 3e x  2 dx, 2 – 2 loge2
4 a cos (2x) –2x sin (2x),

3
loge 2 2x
sin(2x) – 2x cos (2x)
4
π
b 4
5 6π
3x2
6 a x3 + 1
b loge (2)
7 a k = –4
4
10
b f(x) = 3x3 – 4x + 3
8 a 0
b 12
c 32
9
3 1
2
3
10 a y = –π x + 3
3π
b 6– 2
Mathematical Methods Integration Revision
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Answers to extended-response questions
a c = a3
8a3
8a3
b decreasing on (0, 27 ) and increasing on ( 27 , )
1
c y = (x – a3), all are parallel
3
4a3
d [– 27 , )
a6
e 10
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