Math - Integration

INTEGRATION: AN OVERVIEW 30/07/2008 FORMULAE NOT FOUND IN MF15

1

1

nn xx dx c

n

+

= ++∫ where 1n ≠ − ( ) ( )

( )

1

1

nn ax b

ax b dx ca n

+++ =

+∫ + where 1n ≠ −

1 1 lnx dx dx x cx

− = =∫ ∫ + ( ) 1 1 1 lnax b dx dx ax b cax b a

−+ = = + ++∫ ∫

( ) ( )( ) 1

ff f

1

nn x

x x dx cn

+⎡ ⎤⎣ ⎦′ = +⎡ ⎤⎣ ⎦ +∫

( )( ) ( )

f ln f

fx

dx x cx

′= +∫

( ) ( ) ( )f ff x xx e dx e c′ = +∫ ( ) ( ) ( )f f1fln

x xx a dx a ca

′ = +∫

x xe dx e c= +∫

Anglo-Chinese Junior College H2 Mathematics 9740: INTEGRATION (REVISION 2008)

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1ax b ax be dx e ca

+ += +∫ 1ln

x xa dx a ca

= +∫

cos sinx dx x c= +∫ sin cosx dx x c= − +∫ 2sec tanx dx x c= +∫ 2cosec cotx dx x c= − +∫ sec tan secx x dx x c= +∫ cosec cot cosecx x dx x c= − +∫

FORMULAE FOUND IN MF15

1

2 2

1 sin xdx caa x

− ⎛ ⎞= +⎜ ⎟⎝ ⎠−

∫ ( )x a< 12 2

1 1 tan xdx ca aa x

− ⎛ ⎞= +⎜ ⎟+ ⎝ ⎠∫

2 2

1 1 ln2

x adx ca x ax a

−⎛ ⎞= +⎜ ⎟+− ⎝ ⎠∫ ( )x a> 2 2

1 1 ln2

a xdx ca a xa x

+⎛ ⎞= +⎜ ⎟−− ⎝ ⎠∫ ( ) x a<

( )tan ln secx dx x c= +∫ 2

x π⎛ ⎞<⎜ ⎟⎝ ⎠

( )sec ln sec tanx dx x x c= + +∫ 2

x π⎛ ⎞<⎜ ⎟⎝ ⎠

( )cot ln sinx dx x c= +∫ ( )0 x π< < ( ) ( )cosec ln cosec cot 0x dx x x c x π= − + + < <∫ TRIGONOMETRIC FORMULAE USED IN INTEGRATION Trigonometric Identities (NOT FOUND IN MF15)

2 2sin cos 1x x+ = 2 2tan 1 secx x+ = 2 21 cot cosecx x+ = Double Angle Formulae (FOUND IN MF15)

sin 2 2sin cosA A= A ( )

( )

2 2

2 2

2 2

cos 2 cos sin1 2cos 1 cos 1 cos 221 1 2sin sin 1 cos 22

A A A

A A

A A

= −

= − ⇒ = +

= − ⇒ = −

A

A

Factor Formulae (FOUND IN MF15)

( ) ( )

( ) ( )

1 1sin sin 2sin cos2 21 1sin sin 2cos sin2 2

P Q P Q P

P Q P Q P

+ = + −

− = + −

Q

Q

( ) ( )

( ) ( )

1 1cos cos 2cos cos2 21 1cos cos 2sin sin2 2

P Q P Q P Q

P Q P Q P

+ = + −

Q− = − + −

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OTHER TECHNIQUES USED IN INTEGRATION Integration by a given Substitution

Integration by Parts dv duu dx uv v ddx dx

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫ ∫ x

SUMMARY (PLANE AREA & VOLUME OF SOLID OF REVOLUTION)

Area under a curve (bounded by the x-axis)

Area under a curve (bounded by the y-axis)

vertical strips horizontal strips y is written as a function of x x is written as a function of y

curve above x-axis − positive area curve to the right of y-axis − positive area curve below x-axis − negative area curve to the left of y-axis − negative area

b

ay dx∫

d

cx dy∫

Volume of solid of revolution (rotated about the x-axis)

Volume of solid of revolution (rotated about the y-axis)

Volume of solid formed when region under the curve bounded by the x-axis is rotated about the x-axis = 2b

a y dxπ ∫

Volume of solid formed when region bounded by the curve and the y-axis is rotated about the y-axis = 2d

c x dyπ ∫

Volume of solid formed when region between two curves is rotated about the x-axis = 2 2

1 2b ba ay dx y dxπ


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π−∫ ∫

Volume of solid formed when region between two curves is rotated about the y-axis = 2 2

1 2d dc cx dy x dyπ π−∫ ∫

a b δx

y

d

c

δy x

a b

y

x

δx

y δy

c

d

x

x

y


EXERCISE

1. Express 2

3( 3

2)

xx x

−+

in partial fractions.

Hence or otherwise, find 2

3 2( 3)

x dxx x

−+∫ . [ACJC 2006]

2. (i) By using the substitution θsin2=x , find the exact value of ( ) 1 2

0

1 4 3

x dx−∫ .

(ii) The region R, bounded by the curves 1x

y = and ( ) 431 2xy −= , the x-axis, the y-

axis and the line is shown in the diagram below. 3.1=x


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y x= −13

4 2( ) y

x=

1

3.1O

y

R

x

(a) Find the area of R, correct to 3 significant figures. (b) Find the volume formed when the shaded region, R, is rotated through 360°

about the x-axis. [HCI 2006]

3. (a) Find (i) x2∫ (ii) xex d3

∫ xex d . x35

(b) Use the substitution u to find the exact value of x= ∫ +−

9

1 d

54x

xxx .

[NYJC 2006]

4. (i) Find 21d x xdx

⎡ ⎤−⎣ ⎦

in

, simplifying your answer.

(ii) Find 4 s 1x x dx−∫ . [AJC 2006]



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15. (a) The graph of , for 02 1y x= + x≤ ≤ , is shown in the diagram. Rectangles, each of

width 1n

, are drawn under the curve.

(i) Show that the total area of all the n rectangles is given by 2

2

8 36

n nn− +

y

xO 11 2 3 3 2 1n n n

n n n n n n− − −

1 .

(ii) By considering the area of the region bounded by the curve 2 1, 1y x= + x =

and the axes, briefly explain why 2

2

8 3 1 46 3

n nn− +

< .

(b) The region R is bounded by the curves 2 yx = , 4 yx = and the line as shown. 16x =

16x =

x

y

Find the volume of solid formed when R is rotated through four right angles about the y-axis. Leave your answer in the form aπ where a is a real constant corrected to three significant figures. [NJC 2006]

ANSWERS 1.

( ) 22

3 2 1 233

x xx xx x

− +⎛ ⎞= − ⎜ ⎟++ ⎝ ⎠ ( )1 22 1ln tan ln 3

23 3xx x C−− − + +

2. (i) 1 33 23π⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠ (ii) (a) 1.37 (b) 4.56

3. (a) (i) 31

3xe +C (ii)

3 331 13 3

x xe x e C− + (b) 4 3π+

4. (i) 2

2

1 2

1

x

x

−

− (ii) 2 1 2 12 sin 1 sinx x x x x C− −+ − − +

5. (b) 420π

R

2yx =

4yx =

O


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