C4 Integration Exam Questions
Section A – General Integration by Substitution, by Parts and by Partial Fractions
[C4 June 2014(R) Q4]
1.
(a) Express
25
in partial fractions.
x  2 x  1
2
(4)
Figure 2
Figure 2 shows a sketch of part of the curve C with equation y 
5
, x > 0.
x   2 x  1
The finite region R is bounded by the curve C, the x-axis, the line with equation x = 1 and the
line with equation x = 4.
This region is shown shaded in Figure 2.
The region R is rotated through 360° about the x-axis.
(b) Use calculus to find the exact volume of the solid of revolution generated, giving your
answer in the form a + bln c, where a, b and c are constants.
(6)
[C4 Jan 2014(I) Q2]
2.
(i) Find

x
x cos  dx
2
(3)
(ii) (a) Express
1
in partial fractions.
x 1  3x 
2
(4)
(b) Hence find, for 0  x 
1
3

1
dx
x 1  3x 
2
(3)
[C4 June 2013 Q1]
3.
(a) Find  x 2e x dx .
(5)
(b) Hence find the exact value of
1 2 x
 0 x e dx .
(2)
[C4 June 2013 Q5]
4.
(a) Use the substitution x = u2, u > 0, to show that

1
dx =
x(2 x  1)
 u(2u 1) du
2
(3)
(b) Hence show that

9
1
1
a
dx = 2ln  
x(2 x  1)
b
where a and b are integers to be determined.
(7)
[C4 June 2013(R) Q3]
5.
Using the substitution u = 2 + √(2x + 1), or other suitable substitutions, find the exact
value of

4
0
1
dx
2   (2 x  1)
giving your answer in the form A + 2ln B, where A is an integer and B is a positive constant.
(8)
[C4 June 2013(R) Q7]
6.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations
x  27sec3 t ,
y  3tan t ,
0≤t≤
(a) Find the gradient of the curve C at the point where t =

6

3
.
(4)
(b) Show that the cartesian equation of C may be written in the form
y  ( x 3  9) 2 ,
2
1
a≤x≤b
stating values of a and b.
(3)
Figure 3
The finite region R which is bounded by the curve C, the x-axis and the line x = 125 is shown
shaded in Figure 3. This region is rotated through 2π radians about the x-axis to form a solid of
revolution.
(c) Use calculus to find the exact value of the volume of the solid of revolution.
(5)
[C4 Jan 2013 Q2]
7.
(a) Use integration to find
 1 ln x dx .
 3
x
(5)
(b) Hence calculate
2
 1 ln x dx .
 3
1 x
(2)
[C4 Jan 2013 Q5]
8.
Figure 2
Figure 2 shows a sketch of part of the curve C with parametric equations
1
x = 1 – t,
y = 2t – 1.
2
The curve crosses the y-axis at the point A and crosses the x-axis at the point B.
(a) Show that A has coordinates (0, 3).
(b) Find the x-coordinate of the point B.
(c) Find an equation of the normal to C at the point A.
(2)
(2)
(5)
The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and the
x-axis.
(d) Use integration to find the exact area of R.
(6)
[C4 Jan 2013 Q6]
9.
Figure 3 shows a sketch of part of the curve with equation y = 1 – 2 cos x, where x is measured
in radians. The curve crosses the x-axis at the point A and at the point B.
(a) Find, in terms of , the x coordinate of the point A and the x coordinate of the point B. (3)
The finite region S enclosed by the curve and the x-axis is shown shaded in Figure 3. The region
S is rotated through 2 radians about the x-axis.
(b) Find, by integration, the exact value of the volume of the solid generated.
(6)
[C4 June 2012 Q1]
10.
f(x) =
1
C
A
B
=
+
+
.
2
x
(3 x  1) 2
x (3 x  1)
(3 x  1)
(a) Find the values of the constants A, B and C.

(b) (i) Hence find  f ( x) dx .

(4)
2

(ii) Find  f ( x) dx , leaving your answer in the form a + ln b, where a and b are
1
constants.
(6)
[C4 Jan 2012 Q2]
11.

(a) Use integration by parts to find  x sin 3 x dx.


(b) Using your answer to part (a), find  x 2 cos 3 x dx.

(3)
(3)
[C4 Jan 2012 Q4]
12.
Figure 1
Figure 1 shows the curve with equation
 2x 
 2
 , x  0.
 3x  4 
The finite region S, shown shaded in Figure 1, is bounded by the curve, the x-axis and the
line x = 2.
The region S is rotated 360° about the x-axis.
y=
Use integration to find the exact value of the volume of the solid generated, giving your answer
in the form k ln a, where k and a are constants.
(5)
[C4 June 2011 Q7]
13.
Figure 3
Figure 3 shows part of the curve C with parametric equations
x = tan ,
y = sin ,
0 <

2
.
1 

The point P lies on C and has coordinates   3,  3  .
2 

(a) Find the value of  at the point P.
(2)
The line l is a normal to C at P. The normal cuts the x-axis at the point Q.
(b) Show that Q has coordinates (k3, 0), giving the value of the constant k.
(6)
The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x = 3 and
the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of
revolution.
(c) Find the volume of the solid of revolution, giving your answer in the form p 3 + q 2,
where p and q are constants.
(7)
[C4 Jan 2011 Q1]
14.

2
Use integration to find the exact value of  x sin 2 x dx .
0
(6)
[C4 Jan 2011 Q6]
15. The curve C has parametric equations
x = ln t,
y = t2 −2,
t > 0.
Find
(a) an equation of the normal to C at the point where t = 3,
(b) a cartesian equation of C.
(6)
(3)
Figure 1
The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line x = ln 2 and the line
x = ln 4. The area R is rotated through 360° about the x-axis.
(c) Use calculus to find the exact volume of the solid generated.
(6)
[C4 June 2010 Q2]
16. Using the substitution u = cos x +1, or otherwise, show that

 2 cos x  1
sin x dx = e(e – 1).
 e
0
(6)
[C4 June 2010 Q6]
17.
(a) Show that f(θ) =
f(θ) = 4 cos2 θ – 3sin2 θ
1
7
+ cos 2θ.
2
2
(3)

2
(b) Hence, using calculus, find the exact value of   f ( ) d .
0
(7)
[C4 June 2010 Q8]
18.
Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank
is 6 m. Water is flowing into the tank at a constant rate of 0.48π m3 min−1. At time t minutes,
the depth of the water in the tank is h metres. There is a tap at a point T at the bottom of the
tank. When the tap is open, water leaves the tank at a rate of 0.6πh m3 min−1.
(a) Show that, t minutes after the tap has been opened,
75
dh
= (4 – 5h).
dt
(5)
When t = 0, h = 0.2
(b) Find the value of t when h = 0.5
(6)
[C4 Jan 2010 Q7]
19.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations
x = 5t 2 − 4, y = t(9 − t 2)
The curve C cuts the x-axis at the points A and B.
(a) Find the x-coordinate at the point A and the x-coordinate at the point B.
(3)
The region R, as shown shaded in Figure 2, is enclosed by the loop of the curve.
(b) Use integration to find the area of R.
(6)
[C4 Jan 2010 Q8]
20. (a)
Using the substitution x = 2 cos u, or otherwise, find the exact value of
2
1

dx .

2
2
1 x  (4  x )
(7)
Figure 3 shows a sketch of part of the curve with equation y =
4
0 < x < 2.
1 ,
x(4  x 2 ) 4
The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with
equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis
to form a solid of revolution.
(b) Using your answer to part (a), find the exact volume of the solid of revolution formed.(3)
[C4 June 2009 Q3]
21.
f(x) =
4  2x
C
A
B
=
+
+
.
(2 x  1)( x  1)( x  3) (2 x  1)
( x  1) ( x  3)
(a) Find the values of the constants A, B and C.

(b) (i) Hence find  f ( x) dx .

(4)
(3)
2

(ii) Find  f ( x) dx in the form ln k, where k is a constant.
0
[C4 June 2009 Q6]

22. (a) Find   (5  x) dx .

(3)
(2)
Figure 3 shows a sketch of the curve with equation
y = (x – 1)√(5 – x),
1x5

(b) (i) Using integration by parts, or otherwise, find  ( x  1)  (5  x) dx .

(4)
5

(ii) Hence find  ( x  1)  (5  x) dx.
1
(2)
[C4 June 2009 Q8]
23.

(a) Using the identity cos 2θ = 1 – 2 sin2 θ , find  sin 2  d .

(2)
Figure 4
Figure 4 shows part of the curve C with parametric equations
x = tan θ,
y = 2 sin 2θ,
0θ<

2
.
1
and the x-axis.
3
This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution.
The finite shaded region S shown in Figure 4 is bounded by C, the line x =
(b) Show that the volume of the solid of revolution formed is given by the integral

6
k  sin 2  d ,
0
where k is a constant.
(5)
(c) Hence find the exact value for this volume, giving your answer in the form p 2 + q√3,
where p and q are constants.
(3)
[C4 Jan 2009 Q2]
24.
Figure 1
3
. The region R is bounded by the curve, the x (1  4 x )
axis, and the lines x = 0 and x = 2, as shown shaded in Figure 1.
Figure 1 shows part of the curve y =
(a) Use integration to find the area of R.
(4)
The region R is rotated 360° about the x-axis.
(b) Use integration to find the exact value of the volume of the solid formed.
(5)
[C4 Jan 2009 Q6]

25. (a) Find  tan 2 x dx .

(2)
1
(b) Use integration by parts to find 
 3 ln x dx .
x
(4)
(c) Use the substitution u = 1 + ex to show that
3x
1
 e
dx = e2x – ex + ln (1 + ex) + k,

x
2
 1 e
where k is a constant.
(7)
[C4 June 2008 Q2]
26.

(a) Use integration by parts to find  xe x dx .

(3)

(b) Hence find  x 2 e x dx .

(3)
[C4 June 2008 Q7]
27.
(a) Express
2
in partial fractions.
4  y2
(3)
(b) Hence obtain the solution of
2 cot x
for which y = 0 at x =

3
dy
= (4 – y2)
dx
, giving your answer in the form sec2 x = g( y).
(8)
[C4 June 2008 Q8]
28.
Figure 3
Figure 3 shows the curve C with parametric equations
x = 8 cos t,
0t
y = 4 sin 2t,

2
.
The point P lies on C and has coordinates (4, 2√3).
(a) Find the value of t at the point P.
(2)
The line l is a normal to C at P.
(b) Show that an equation for l is y = –x√3 + 6√3.
(6)
The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded
in Figure 3.

2
(c) Show that the area of R is given by the integral  64 sin 2 t cos t dt .

(4)
3
(d) Use this integral to find the area of R, giving your answer in the form a + b√3, where a
and b are constants to be determined.
(4)
[C4 Jan 2008 Q3]
29.
1
. The finite region bounded by the curve,
(2 x  1)
the x-axis and the lines x = a and x = b is shown shaded in Figure 2. This region is rotated
through 360° about the x-axis to generate a solid of revolution.
The curve shown in Figure 2 has equation y =
Find the volume of the solid generated. Express your answer as a single simplified fraction, in terms
of a and b.
(5)
[C4 Jan 2008 Q4]
  x
30. (i) Find  ln   dx .
 2
(4)

2
(ii) Find the exact value of  sin 2 x dx .

(5)
4
[C4 Jan 2008 Q7]
31.
Figure 3
The curve C has parametric equations
1
, t > 1.
(t  1)
The finite region R between the curve C and the x-axis, bounded by the lines with equations
x = ln 2 and x = ln 4, is shown shaded in Figure 3.
x = ln (t + 2),
y=
(a) Show that the area of R is given by the integral
2
1

dt .

0 (t  1)(t  2)
(4)
(b) Hence find an exact value for this area.
(6)
(c) Find a cartesian equation of the curve C, in the form y = f(x).
(4)
(d) State the domain of values for x for this curve.
(1)
[C4 June 2007 Q2]
32. Use the substitution u = 2x to find the exact value of
1
2x

 (2 x  1) 2 dx.
0
(6)
[C4 June 2007 Q3]

33. (a) Find  x cos 2 x dx .

(4)

(b) Hence, using the identity cos 2x = 2 cos2 x – 1, deduce  x cos 2 x dx .

(3)
[C4 June 2007 Q4]
34.
C
B
2(4 x 2  1)
A+
+
.
(2 x  1)
(2 x  1)
(2 x  1)( 2 x  1)
(a) Find the values of the constants A, B and C.
(4)
2
2(4 x 2  1)

(b) Hence show that the exact value of 
dx is 2 + ln k, giving the
1 (2 x  1)( 2 x  1)
value of the constant k.
(6)
[C4 Jan 2007 Q2]
35.
Figure 1
y
–
The curve with equation y =
x
0
1
, x > – 12 , is shown in Figure 1.
3(1  2 x)
The region bounded by the lines x = – 14 , x =
Figure 1.
1
2
, the x-axis and the curve is shown shaded in
This region is rotated through 360 degrees about the x-axis.
(a) Use calculus to find the exact value of the volume of the solid generated.
(5)
Figure 2
A
B
Figure 2 shows a paperweight with axis of symmetry AB where AB = 3 cm. A is a point on the
top surface of the paperweight, and B is a point on the base of the paperweight. The paperweight
is geometrically similar to the solid in part (a).
(b) Find the volume of this paperweight.
(2)
[C4 June 2006 Q3]
36.
Figure 3
y
2
O
x
x
, 0  x  2, is shown in Figure 1. The finite region
2
enclosed by the curve and the x-axis is shaded.
The curve with equation y = 3 sin
(a) Find, by integration, the area of the shaded region.
(3)
This region is rotated through 2 radians about the x-axis.
(b) Find the volume of the solid generated.
(6)
[C4 Jan 2007 Q3]
37. Using the substitution u2 = 2x – 1, or otherwise, find the exact value of
5
3x

dx .

1  (2 x  1)
(8)
[C4 Jan 2006 Q4]
38.
Figure 1
y
y = xex
R
O
1
3
x
Figure 1 shows the finite region R, which is bounded by the curve y = xex, the line x = 1, the
line x = 3 and the x-axis.
The region R is rotated through 360 degrees about the x-axis.
Use integration by parts to find an exact value for the volume of the solid generated.
(8)
[C4 Jan 2006 Q8]
39.
Figure 2
y
R
O
x
The curve shown in Figure 2 has parametric equations
x = t – 2 sin t,
0  t  2.
y = 1 – 2 cos t,
(a) Show that the curve crosses the x-axis where t =

3
and t =
5
.
3
(2)
The finite region R is enclosed by the curve and the x-axis, as shown shaded in Figure 2.
(b) Show that the area R is given by the integral
5
3
2
 (1  2 cos t ) dt .
3
(3)
(c) Use this integral to find the exact value of the shaded area.
(7)
[C4 June 2005 Q3]
40.
(a) Express
5x  3
in partial fractions.
(2 x  3)( x  2)
(3)
5x  3

(b) Hence find the exact value of 
dx , giving your answer as a single
2 (2 x  3)( x  2)
logarithm.
(5)
6
[C4 June 2005 Q4]
1
41.
Use the substitution x = sin  to find the exact value of
2
1
dx .
3

0 (1  x 2 ) 2
(7)
Solutions
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Question 7
Question 8
Question 9
Question 10
Question 11
Question 12
Question 13
Question 14
Question 15
Question 16
Question 17
Question 18
Question 19
Question 20
Question 21
Question 22
Question 23
Question 24
Question 25
Question 26
Question 27
Question 28
Question 29
Question 30
Question 31
Question 32
Question 33
Question 34
Question 35
Question 36
Question 37
Question 38
Question 39
Question 40
Question 41