First year:
(I)
Basic integration
Integration: areas
The symbol
b
 f ( x)dx
a
means “area of the set situated between the graph of f , the x  axis, the line
of equation x  a and the line of equation x  b ”.
It’s easier to understand this notion through an example:
2
 ( x  1)(3  x)dx
is the area of the set situated between the x  axis, the graph
1
of f and the vertical lines of equation x  1, x  2 .
(II)
Basic Rules
Integration is, intuitively, the opposite of differentiation. Mathematically, we
say that if there exists a function F such that
d
F ( x)  f ( x)
dx
b
then
 f ( x)dx  F (b)  F (a) .
a
The result above is sometimes called the fundamental theorem of calculus.
When you are asked to integrate a function f , the trick is to find this function
F.
If such a function F exist, then instead of saying “a function F such that
d
F ( x)  f ( x) ”, we usually say
dx
F ( x)   f ( x)dx
Let’s have a look at an example:
2
Calculate:
 2xdx .
1
The first step is to find
 xdx .
We know that the derivative of x 2 is 2x , so that we have
 2 xdx  x
It remains now to calculate the actual value of the integral:
2
 2 xdx  [ x
]  2 2  1 2  3.
2 2
1
1
So far so good? Now try to calculate the following integral:
2
 xdx .
1
Here are a few examples:
-
3
 x dx 
x4
4
3
-
 5dx  [5x]
3
0
 5  3  5  0  15.
0
-
2 7
2  x8  x8
 5 x dx  5  8   20 .
The general rule is
n
 x dx 
x n1
.
n 1
2
.
(III)
Exercises
Find the following integrals:
-
 x dx
-
 3x dx
2
4
3
-
x
1
3
dx