---

Main Points Introduction Basic Properties of Indefinite Integrals Basic Properties of Definite Integrals Standard Forms Use of Trigonometry Identities By Substitution By Parts

Introduction

• Integration may be thought of as the inverse of differentiation.
• Suppose � and F are two functions related by

d ¾ dx F(x)  = �(x), then  ò �(x) dx  = F(x) + C.

• We call

• The process of finding ò �(x) dx for a given function � is called integration.

• For a well-behaved function �,

• We call

ób  �(x) dx õa	- the definite integral of � with respect to x on [a, b],
a	- the lower limit,
b	- the upper limit.

Basic Properties of Indefinite Integrals

1.	d ¾ dx [ò �(x) dx]	= �(x)
2.	ò( d ¾ dx �(x)) dx	= �(x) + C
3.	ò dx = ò 1 dx	= x + C
4.	ò k�(x) dx	= kò �(x) dx    where k is a constant
5.	ò [�(x) ± g(x)]dx	= ò �(x) dx ± ò g(x) dx

Basic Properties of Definite Integrals

1.	óa  �(x) dx õa	= 0
2.	ób  �(x) dx õa	= - óa  �(x) dx õb
3.	ób  �(x) dx õa	=  óm  �(x) dx +  ób  �(x) dx õa õm

Standard Forms

1.	For n ¹ 1,
(a)	ò xn dx	xn + 1    =  ¾¾¾  + C   n + 1
(b)	ò (ax + b)n dx	(ax + b)n    =  ¾¾¾¾  + C   a(n + 1)
(c)	ò �'(x)[�(x)]n dx	[�(x)]n + 1    =  ¾¾¾¾¾  + C   n + 1

2. (a)	ó ô õ 1 ¾ x  dx	= ln |x| + C
(b)	ó ô õ 1 ¾¾¾ ax + b  dx	=  1 ¾ a ln |ax + b| + C
(c)	ó ô õ �'(x) ¾¾¾ �(x)  dx	= ln |�(x)| + C

3. (a) (i)	ò cos x dx	= sin x + C
(ii)	ò sin x dx	= - cos x + C
(iii)	ò sec2 x dx	= tan x + C
(iv)	ò cosec2 x dx	= - cot x + C
(v)	ò sec x tan x dx	= sec x + C
(vi)	ò cosec x cot x dx	= - cot x + C
(b)	ò cos (ax + b) dx	=  1 ¾ a sin (ax + b) + C
	similarly for the rest
(c)	ò �'(x) cos �(x) dx	= sin �(x) + C
	similarly for the rest

 

4. (a)	ò ex dx	= ex + C
(b)	ò eax + b dx	=  1 ¾ a eax + b + C
(c)	ò �'(x) e�(x) dx	= e�(x) + C

5. (a) (i)	ó ô õ 1 ¾¾¾¾ Ö(1 - x2)  dx	= sin -1 x + C
(ii)	ó ô õ 1 ¾¾¾¾ Ö(a2 - x2)  dx	= sin -1  x ¾ a  + C
(iii)	ó ô õ �'(x) ¾¾¾¾¾¾¾ Ö(a2 - [�(x)]2)  dx	= sin -1  �(x) ¾¾ a  + C
(b) (i)	ó ô õ 1 ¾¾¾ 1 + x2  dx	= tan-1 x + C
(ii)	ó ô õ 1 ¾¾¾ a2 + x2  dx	=  1 ¾ a tan-1  x ¾ a  + C
(iii)	ó ô õ �'(x) ¾¾¾¾¾ a2 + [�(x)]2  dx	=  1 ¾ a tan-1 �(x) ¾¾ a  + C
(c) (i)	ó ô õ 1 ¾¾¾ a2 - x2  dx	=  1 ¾ 2a ln æ ç è a + x ¾¾¾ a - x ö ÷ ø  + C	for |x| < a
(ii)	ó ô õ 1 ¾¾¾ x2 - a2  dx	=  1 ¾ 2a ln  æ ç è x - a ¾¾¾ x + a ö ÷ ø  + C	for |x| > a

Use of Trigonometry Identities

  1.  ò sin mx cos nx dx , ò sin mx sin nx dx or ò cos mx cos nx dx

Use the factor formulae to express the integrand as the sum or difference of 2 sines or cosines and then integrate.

sin P cos Q	=	½[sin (P + Q) + sin (P - Q)]
cos P cos Q	=	½[cos (P + Q) + cos (P - Q)]
sin P sin Q	=	-½[cos (P + Q) - cos (P - Q)]

  2.  ò sinⁿ x dx or ò cosⁿ x dx

(a)  When n is even

Use

cos2 x	= ½(1 + cos 2x)
sin2 x	= ½(1 - cos 2x)

repeatedly to express the integrand as cosines of multiple angles.

(b)  When n is odd

Keep one of the term and express the rest into the complementary function using

sin² x + cos² x = 1.

Eg.	sin3 x	= sin2 x sin x
		= (1 - cos2 x) sin x
	cos5 x	= cos4 x cos x
		= (1 - sin2 x)2 cos x
		= (1 - 2 sin2 x + sin4 x) cos x

By Substitution

By Parts

L  - Logarithmic Functions I  - Inverse Trigonometric Functions A  - Algebraic Functions T  - Trigonometric Functions E  - Exponential Functions