JDM Chap 1 Integration

Unit Matematik

Kolej Matrikulasi Johor

Kementerian Pendidikan Malaysia
Jom –Do- Maths (JDM)

Chapter 1 Integration

1. Find

(a)  1 (4  3x4 ) dx
x2

(b)  x2  1dx
x

(c)   2  x  2
 x 
dx

(d)  34x  55 dx

(e)  2 dx
 5x)3
(4

(f)  3 3 dx
10 
2x

2. Find

(a) 1  e3x dx
e2x

 (b) 2
ex  ex
dx

(c) 423xdx

  (d) 32x1 52x1 dx

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3. Find

(a)  cos 4x  5sec2 2x dx

(b)  5sin  3x  5cos ec2 4x dx

(c)  cos x dx
sin 2 x

(d)  cos 3x cos 4x  sin 4x sin 3x dx

(e)  cos x  sin 3x   sin x  cos 3x  dx
 2   2   2   2 

(f)  cos2 3x dx

(g)  sin 2  3x  dx
 2 

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Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia

Jom-Do Maths
Chapter 1 Integration (Set 2)

1. Find (b)  cos 3x  sin x  dx
 4  4 
(a)  sin 3x sin 5xdx
(c)  cos4x  2cosx 1 dx (d)  sin3x 1cosx  2 dx

2. By using integration by substitution method , find

(a)  3x8 1 dx (b) x2e14x3 dx
x9  3x

(c)  3e x dx  (d) e3x dx
e3x  2 3
x

(e)  6x5 dx (f)  x2 dx
2  x3 x4

(g)  1  ln x dx (h)  ln 4x dx
5x x

(i)  ln x 1 dx (j) 2 x ln x dx
x ln x 1 ln x

(k)  cos x dx (l)  sec2 x dx
sin x tan5 x

(m)  cos3 3x dx (n)  sin3 4x cos 4x dx

(o)  sin3 2x cos2 2x dx

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Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia

JOM- DO- MATHS (JDM)
Chapter 1 Integration (Set 3)

By using integration by parts , find

(a)  (2  x) exdx (b) sec2 (3x 1) etan(3x1)dx

(c)  x ln x dx (d)  ln x dx
x4

(e)  ln x dx (f)  x sec2 x dx
x (h)  x x  3 dx

 (g)  x 3x dx

(i)  xsin x cos x dx (j) x2e2x1 dx

 2 (l)  x sin2 2x dx

(k)  x ln x

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Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia

JOM-DO-MATHS ( JDM)
Chapter 1 Integration (Set 4)

1. Using partial fraction, find  3x2  2x  2 dx
(x 1)(x2  2)

2. Given that x2  P  Q  R . Find the values of P, Q and R. Hence, show that
x2 16 x4 x4

 x2 dx  x  2ln x4 C .
x2 16 x4

3. By using partial fraction , show that

x 1  1
x2 1 2(x 1) 2(x 1)

Hence, evaluate 5 x dx
2 1
 x

2

4. Show 3x 3 10x2 6  3x  1   6  3x  . Hence, solve   3x3 10x2  6 dx
3x  x2   3x  x2
3 xx

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5. (a) Use the long division to show that x4  2x3  4x2  x  3  A(x)  8x 1 ,
x2  x  2 x2  x  2

where A(x) is a function in x.

(b) Find the values of B and C if 8x 1  B  C .
x2  x  2 (x  2) (x 1)

(c) Hence, evaluate 3 x4  2x3  4x2  x  3 dx .

0 x2  x  2

6. Express 3x2  7x  6 in the form of partial fraction.Hence, evaluate
(x  3)2 (x 1)

2 3x2  7x  6 dx . Give your answer in the form of a  ln b .

1 (x  3)2 (x 1)

7. Given 1  A  B  C . Find the values of A, B and C .Show that
x2 (1  x) x x2 1  x

x3  x2 1  1 1 x3  x2 1 31
2 x2 (1 x) 42
x2 (1 x)
1 . Hence, prove that dx  ln .
x2 (1 x)

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Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia

JOM-DO-MATHS (JDM)
Chapter 1 Integration (Set 5)


3

1. Find  sin 2 cos2  d

0

2

2. Evaluate  x3 x 4  5dx

1

 3. Show that e x ln x dx  1 1 e2 .
14

2

4. Find the exact value of  t3 t 2  1 dt

1

5. Show that

4 ln xdx  8 ln 2  4
x


1

6. Given 8x3 14x2  6  Qx  A  B . Find Q (x), A and B. Hence, find the
2x2  3x x 2x  3

value of 2 8x3  14x2  6 dx .
1 2x2  3x

7. Let R be the region bounded by y  x ln x, y  0, x  1 and x  4 . Find
(a) The area of R,
(b) The volume of revolution when R is rotated through 360o about the x- axis

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8. Find the area of the region R bounded by the following graphs.

y y b=)x b) y
a)

R x R
4 y  x  x2

y y
c) y  x2
d) 64 y  x3
y4
R R

4x 8
x

e) f)

x

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9.

The diagram shows the graphs of y  x2 and y  2  x 2 . Calculate

a) the shaded area

b) the volume of the solid generated when the shaded area is rotated through 

radians about the yaxis.

10. Given the curve y  4x2 and the line y  6x
(a) Find the intersection points.
(b) Sketch the region enclosed by the curve and the line.
(c) Calculate the area of the region enclosed by the curve and the line.

Calculate the volume of the solid generated when the region is revolved completely about
the y-axis

11. Given the curve y2  x and the line y  2x 1.
(a) Determine the points of intersection between the curve and the line.
(b) Sketch the curve and the line on the same axes. Shade the region R
bounded by the curve and the line. Label the points of intersection.
(c) Find the area of the region R.

Calculate the volume of the solid generated when the region R is rotated 2 radians about
the y-axis

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12.

The figure shows part of the curve y2  a(a  x) , where a is a constant,
and R is the region bounded by the curve and the coordinate axes.
(a) Find the area of the region R.
(b) Calculate the volume V1 , of the solid generated when R is rotated 360

about the x-axis.
(c) If V2 is the volume of the solid generated when R is rotated 360 about the

y-axis, show that V2 :V1  16 :15 .

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