MATHEMATICS EXTENSION 2
4 UNIT MATHEMATICS
TOPIC 4: INTEGRATION
4.1
THE INDEFINITE INTEGRAL
Differentiation is concerned with the rates of change of physical quantities. It is a
fundamental topic in mathematics, physics, chemistry, engineering etc. Consider a
continuous and single value function y  f ( x ) . The rate of change of y with respect to x
at the point x1 is called the derivative and equals the slope of the tangent to the curve
y  f ( x ) at the point x1. The process of finding the derivative of a function is called
differentiation.
The reverse problem is also an extremely important part of mathematics – given a
function f(x) which represents the rate of change of some quantity with respect to x,
what is the function F(x) such that
f ( x) 
d F ( x)
dx
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The process of working back from the derivative of F(x) to the function F(x) is called
integration.
F ( x )   f ( x ) dx
indefinite integral
Integration is the inverse process of differentiation. Whereas there are definite rules
for the differentiation of any function, there are no such rules for integration.
Given that
f ( x) 
d F ( x)
dx
then F(x) called an indefinite integral because the value F(x) depends upon some
arbitrary constant C.
F ( x )   f ( x ) dx  C
C is a constant
Example
y  3x 2  2 x  15
z  3x 2  2 x  99
dy / dx  6 x  2 dz / dx  6 x  2
y  z dy / dx  dz / dx
F ( x )    dy / dx  dx    dz / dx  dx    6 x  2  dx
F ( x )  3x 2  2 x  C
The value of C can’t be determined from dy / dx or dz / dx alone.
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The value of an integral is unaffected by multiplication by a constant
 a f ( x) dx  a  f ( x) dx
a is a constant
The integral of a sum can be found by integrating each term separately
  f ( x)  g ( x)  dx   f ( x) dx   g ( x) dx
STANDARD INTEGRALS
Some integrals are of a standard form and can be integrated relatively easily and
others can be converted to a standard form using a variety of different techniques.
Many integrals can’t be integrated analytically at all, for example,  exp   x 2  dx .
However, the integral can be evaluated using numerical techniques such as Simpson’s
rule.
n
 x dx 
1
x n 1
C
n 1
 x dx   x
1
n  1 x  0
dx  log e  x   C
1
 sin  a x  dx   a cos  a x   C
log e ( x )  ln( x )
x0
a0
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1
 cos  a x  dx  a sin  a x   C
1
 sec  a x  dx  a tan  a x   C
2
a0
a0
1
 cosec  a x  dx   a cot  a x   C
2
a0
1
 sec  a x  tan  a x  dx  a sec  a x   C
1
a0
 cosec  a x  cot  a x  dx   a cosec  a x   C
e
dx 
ax
1 ax
e C
a
a0
ax
 a dx  log e  a   C
x

a 1 a  0
 x
 x
dx  sin 1    C   cos1    C  a  x  a a  0
a
a
a x
1
2
2
1

x a

x a
a
a0
2
2



dx  log e x  x 2  a 2  C
2
dx  log e x  x 2  a 2  C
1
2

2
1
1
 x
dx  tan 1    C
2
x
a
a
xa0
a0
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