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Integration
Integration is the reverse of differentiation. Given the derivative of a function, the process of finding the
original function is the reverse of differentiation, f is said to be an antiderivative of
/ (^) f. Given
/
f x ^ , then^ f^ x ^4 x ^ c.
Example 1
Find the antiderivative of any constant of 0
/ f x .
Solution:
The derivative of any constant function is zero. Therefore, the antiderivative is f x c.
Example 2
Find the antiderivative of 2 7
/ (^) f x x .
Solution:
The antiderivative is f x x 7 x c
2
Definition
If F x is a function of x such that F x f ( x )
dx
d
, then we define integral of f x with respect to
(w.r.t) x to be the function F x and we write
f^ xdx^ F ^ x
Thus F x f x f xdx F x dx
d
For example
5 4 x 5 x dx
d , we have
4 5 (^) 5 x dx x.
Moreover, if c is any constant, then
5 4 x c 5 x dx
d .
So, in general, x dx x c
4 5
NOTE: Different values of c will give different integrals and thus integral of a function is unique.
Thus in general if f is a continuous function, then
f xdx F x c
If F x f x
/ , where c is the constant of integration.
Rules of Integration
Rule 1. Constant functions
kdx kx c , where k is any constant.
Example
Evaluate the following integrals
(i) 7 dx
(ii) dx 3
(iii) 3 dx
(iv)
dx 7
(v) 0 dx
(vi) dx 2
Solution:
(i) ^7 dx^ ^7 x c
(ii) dx x c 3
(iii) 3 dx 3 x c
(iv) dx x c 7
(v) 0 dx 0 x c c
(vi) dx x c 2
Rule 2: Power Rule
c n
x xdx
n n
1
1 , provided that n 1.
Example
Evaluate the following integrals
(i) xdx
(ii) xdx
2
(iv)
dx x
(v) xdx
5 20
(vi)
xdx
3 5
(vii) dx x
(^5 )
Solution:
(i) c x c
x xdx^
2
2 2 2
(ii) c
x c
x dx
x
5 5 3 15
2 3 3
(iii) c x c x c
x xdx
2
3 2
(^23)
3
3 2 3
(iv) c x c x c
x dx x dx x
^
2
1 2
(^21)
1 2
1 5 10 1
(v) x dx x c x c
5 6 6
3
(vi) xdx x dx x c x c x c x c
3
4 3
4 3 1 4 3
1 3 3 1
4
(vii) c x c
x dx x dx x
^
5
(^51)
1 5
4
(^5 )
Rule 4: Addition or Difference
If (^) f x dx and (^) g x dx exist, then f x g x dx f xdx g x dx
Example
Evaluate the following integrals.
(i) 3 x 5 dx
(ii) x 5 x 4 dx
2
(iii) x dx x
x
5
(iv) x dx x
x
3
(v) dx x
x x x (^)
4 3 3
Solution:
(i) ^ x dx^ xdx dx x c ^ x c x 5 x c 2
2 1 2
2
NOTE: Even though the two integrals technically results in separate constants of integration, these constants may be considered together as one.
(ii) x c
x x x x dx
3 2 2
(iii) x x c x
x dx x x
x (^)
^5 2
2 3
6 4
5
(iv) x dx x x x x c x
x (^)
^13 2
3
(v) x x c
x dx x x dx x
x x x
3
4 3 2
4 3
Rule 5: Power Rule Exception
dx x dx x c x
^
ln
Rule 6: Exponential Function
c a
e e dx
ax ax ^ , where^ a is a constant.
Example
Evaluate the following integrals.
(i) edx
x
(ii) e dx
x
(iii) e^ dx
x
3 4
(iv) e dx
x x
2 4
(v) dx x x
x x x
3 3
(vi) dx x
x x x
2
3 2 2 3 4 5
(vii) 2 3 x 1 4 x 7 2 x dx
(ii) dx x
x (^)
(iii) 40 xdx
(iv) x x dx
2 9 7 13
(v) e dx
x
6
(vi) dx e
2 x
(vii) dx
x
(^)
3 2
(viii) dx x
x x (^)
(ix) dx e
e e x
x x
5 8
3
(x) dx x
x
3
2
2 1
Methods of integration
1. Integration by Substitution.
By suitable substitution, the variable x in f x dx
is changed into variable u so that the
integrand f x is changed into F u which is easily integrable.
For example, to evaluate dx x x
x
2 , we notice that 7 13 2 7
2 x x x dx
d .
Therefore, we put 7 13
2 u x x , then 2 x 7 dx
du
or du 2 x 7 dx
du u c x x c u
dx x x
x
ln ln 7 13
2
Note: There is no rule for finding a proper substitute and the best guide in the matter is
experience. However, the following suggestions will be useful.
(i) If the integrand is of the form f ax b
/ , then we put u ax b and a dx
du or
du adx or du a
dx
1 .
Thus
c a
f ax b c a
f u f udu a a
du f ax bdx f u
^
/ /^1 /
(ii) When the integrand is of the form
n n x f x
1 / .
We put
n u^ ^ x and
1
n nx dx
du or du^ nx dx
n 1 .
Thus f x c n
f u c n
f udu n n
du x f x dx f u
n n n
1 / /^1 /^11
(iii) When the integrand is of the form f x f x
n (^) /
..
We put u f x and du f x dx
/
Thus
c n
f x c n
u f x f x u du
n n n n
1 1
1 1 /
(iv) When the integrand is of the form
f x
f x
/ .
We put u f x and du f x dx
/ .
Thus
u c f x c
u
du dx f x
f x ^ ln ln
/
(v) When the integrand is of the form
f x f x e
/ .
We put u f x and du f x dx
/
Thus
f xe dx e du e c e c
f x u u fx ^
/
Example
c
u u du
u
du dx
x
x dx x
x
1 5
(^151) 5
1
5
1 5 3 1
2
5 3
2
c
u
4
5
4
u
3 x 7 c 36
(iv) x x ^3 dx
4 5 1 ^1
Let
5 u 1 x , then
4 5 x dx
du or x dx
du (^) 4
5
c u c x c
u u du
du x x dx u
^ ^3
5 4 3
(^34)
4 3
1 3
1 3 4 5 1 1 20
(v) xe dx
x
2 2
Let
2 u^ ^ x , then x dx
du 2 or du^ ^2 xdx
c e c
e xe dx edu
x
u x u ^
2 2
1
(vi) x e dx
2 3 x^3
Let
3 u 3 x , then
2 9 x dx
du or x dx
du (^) 2
9
x e dx e du e c e c
x u u x ^
2 33 33
9
1
9
1
9
1
(vii) x x dx x x ^2 dx
2 3 2 3 1 ^5 ^5
Let^5
3 u ^ x , then
2 3 x dx
du or x dx
du (^) 2
3
c u c x c
u du
u x x dx x x dx
^ ^2
3 3 2
(^23)
3 2
1 2 2 3 2 3 1 5 9
(viii) x x dx x x dx
2
1 2 2 2 1 2 1
Let
2 u 1 x , then x dx
du 2 or du 2 xdx
x^ ^ x dx x ^ x ^ dx u du u c ^ x ^2 ^ c
2 3 2
3 2 2 2 12 1 1 3
2
3
2 2 1 2 1
(ix)
dx x
dx x
1 2 1
Let u 2 x 1 , then 2 dx
du or dx
du 2
c u c
u u du
du
u
dx x
dx x
2
(^21)
1 2
1
2
1 2
1 2
x ^2 c
1 2 1
(x) e e dx e (^) e dx
x x x x 1 ^ ^1
Let
x u 1 e , then
x e dx
du or du e dx
x
e e dx e e dx u du u c e c
x x x x x ^ ^2
3 2
3 2
1 2
1 1 3
Exercise
Evaluate
(i) dx x
x
ln
(ii) x e dx
x
3 4 4
(iii)
x e dx
x x
2 2 1
(iv) x x dx
2 4 3
(v) x x dx
2 1
(vi) x x^2 dx
3 ^1
(vii) x x dx
3 4 2 ^2
(viii) e e dx
x x
3 1
(ix) xe dx
x
(^2) 1
(x)
dx x
x
3 4
2
(xi) dx e
dx x 1
(xii)
dx x x
1 ln
(xiii) e e e e dx
x x x x
2
xe dx xe e dx xe e c
x x x x x ^
(ii) xdx ln
udv uv vdu
Let u ln x , dx x
du 1 or x
dx du and
let dv dx , on integrating both sides
dv ^ dx
i.e. v x
x
dx ln xdx x ln x x
x ln x dx
x ln x x c
(iii) xe dx
x
3
udv ^ uv vdu
Let u x , then 1 dx
du or du dx and
Let dv e dx
3 x , on integrating we get
i.e. dv e dx
x ^
3
3 x e v
dx
xe e xe dx
x x x
3 3 3
e dx
xe (^) x
x
3
3
c
xe e
x x
3 3
c
xe e
x x 3 9
3 3
(vi) x dx
2 ln
udv ^ uv vdu
Let
2 u ln x , then x dx x
du ln
or xdx x
du ln
and
Let dv dx , on integrating we get i.e. dv (^) dx
v x
x dx
x
x dx x x x
^ ln
ln ln
2 2
x ln x 2 ln xdx
2 ……….(1)
Again applying integration by parts for
2 ln xdx
udv ^ uv vdu
Let u ln x , then dx x
du 1 or x
dx du and let dv 2 dx , on integrating both sides
we get v 2 x
x
dx 2 ln xdx 2 x ln x 2 x
2 x ln x 2 dx
2 x ln x 2 x c ………….. (2)
The substituting the right hand side of equation (2) into equation (1) gives
ln^ x^ ^ dx ^ x ^ ln x ^2 ln xdx
2 2
x ln x 2 x ln x 2 x c
2
x ln x 2 x ln x 2 x c
2
Exercise
Evaluate the following integrals
(i) xe dx
x
2 Ans. xe e c
x x
2 2
4
(ii) x e dx
x
2
Ans. e x c
x
1
(iii) xe dx
x
Ans. e x x c
x 2 2
2
(iv) x e dx
x
3 Ans. x x x e c
x 3 6 6
3 2
(v) x ln xdx
2 Ans. c
x x
x 9
ln 3
3 3
(vi) dx x
x
ln
Ans. 2 x^2 ln x 2 c
1
Putting x 3 , we get 4
A
4 1
1
4 3
9
2 3
2 3 2 x
dx
x
dx dx x x
x
^ x ^ ln^ x 1 c
ln 3 4
(iii) Let
2
x
B
x
A
x x x x
x x
Ax Bx
1 A x 2 B x 3
Putting x 2 , we get 5
B .
Putting x 3 , we get 5
A
5 2
1
5 3
1
6
1 2 x
dx
x
dx dx x x
x ln x 2 c
ln 3 5
(iv) On dividing, we get
x x
x
x x
x x
2 2
2 1 4 7
Let
x
B
x
A
xx
x
x x
Ax Bx
1 4 x A x 1 Bx
Putting x 1 , we get B 5
Putting x 0 , we get A 1
dx xx
x dx x x
x x
2
2
dx x
dx x
dx
1
1 5
1 7
7 x ln x 5 ln^ x 1 c
Exercise
Evaluate the following integrals
(i)
dx x x
x
(ii)
dx x x
x 1 1
(iii) dx x x
2 1
1 2
(iv) 2 3 1
2 t t
dt
(v) dx x x
x x
2
2
dx
x x
1
2
2 2 2
1
2
3 2 2 3 4
x
x x
3 2 3 2
Definite Integral
Let f x and g x be two functions of x such that
f^ xdx^ g ^ x
Then, the definite integral of f x over the interval a , b , denoted by f x dx
b
a
(^) is defined as
f xdx g x g b g a
b a
b
a
Where a and b are two real numbers, and are called, respectively, the lower and the upper
limits of the integral.
Example
Evaluate
(i) x dx
1
0
2 1
(ii) x dx
2
0
(iii) dx x
4
2
(iv) e x dx
x
2
1
3 2 3
Solution:
(i)
1
0
(^13)
0
2
3
x
x x dx
3 3
(ii) x dx
2
0
Let u 6 x 4 , then 6 dx
du or 6
du dx
2
0
2
3
2
0
2
3
2
0
2 2 3
0
2
2 1
0
2
^ u u
u u du
du x dx u
2
0
2
3 6 4 9
x
^2
3 2
3 6 0 4 9
^2
3 2
3 4 9
(iii) ln ln 4 ln 2 ln 2 ln 2 2 ln 2 ln 2 ln 2
(^1 ) 2
4
2
dx x ^ x
(iv) 7 3
6 3
(^26363)
1
3
(^23)
1
3 2
e e
e e e e x
e e x dx
x x
Exercise
1. Find the area bounded by the graphs of f x x x
2
, g x 2 x , for 2 x 3.
2. Find the area bounded by the graphs of
2
f x 2 x , g x 4 2 x , for 2 x 2.
3. Find the area bounded by the graph of 9
2
f x x and the x axis over the indicated
intervals
(i) 0 , 2
(ii) 2 , 4
