### 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
 ò �(x) dx - the indefinite integral of � with respect to x, C - the constant of integration, � - the integrand, ò - the integral sign, F - an anti-derivative of �.
• The process of finding ò �(x) dx for a given function � is called integration.

• For a well-behaved function �,
 ób �(x) dx õa
 = [F(x)] b a
= F(b) - F(a),

where F is an anti-derivative of �.
• 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

3.  ò tann x dx
Use the identities 1 + tan2 x = sec2 x to rewrite tann x.

### By Substitution

Integrand contains
Use the substitution
(ax + b)n
u = ax + b
(a2 - x2)n/2
x = a sin q
(a2 + x2)n/2
x = a tan q
cos2n + 1 x
s = sin x
sin2n + 1 x
c = cos x
 1 ¾¾¾¾¾¾¾¾ a + b cos x + c sin x
t = tan (x/2)

### By Parts

 ó õ u dv ¾ dx dx = uv - ó õ v du ¾ dx dx

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

Hosted by www.Geocities.ws