Class notes Mathematics Higher Engineering Mathematics

KOLHAPUR INSTITUTE OF TECHNOLOGY’S,
COLLEGE OF ENGINEERING (AUTONOMOUS), KOLHAPUR


DEPARTMENT OF BASIC SCIENCES AND HUMANITIES
Faculty of Mathematics
First Year B.Tech. (SEM - II) 2020-2021
ENGINEERING MATHEMATICS- II (UBSH0201)

Unit 3: Special Functions

In this topic we define and study properties of two special functions namely Gamma
function and Beta function. These functions play important role in applied mathematics
especially for evaluation of improper integrals.
Properties of Definite Integration (Revision)
b a a a
1.  f ( x)dx    f ( x)dx 2.  f ( x)dx   f (a  x)dx
a b 0 0
a a/2 a/2 a a
3.  f ( x)dx   f ( x)dx   f (a  x)dx 4.  f ( x)dx  2  f ( x)dx if f(x) is even
0 0 0 a 0
=0 if f(x) is odd
1. Gamma Function:

Definition: For any real number n > 0, the gamma function n is defined as,


n   e  x x n 1dx
0

x n
Gamma Function may also be remembered as e x dx  n  1
0

For e.g .  e  x x 4 dx  5
0

Properties:

1
1. 1  1 2. 0   3.  
2

4. n  (n  1)! if n is positive integer and e.g. 5  (5  1)! 4! 24



Faculty of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur Page 1

, 7 5 5 5 3 3 5 3 1 1 15
5. n  (n  1) n  1 Or n 1 n n e.g.     
2 2 2 22 2 222 2 8

n 1
6. for negative fraction n we use, n 
n
5  3 2  3  3 1 9 1
e.g.           
3  5  3  5   2  3 10 3

 1 1 3
7. n 1  n  For 0 < n < 1 e.g. If n  then  2
sin n 4 4 4

1 1 2 2 1 1 5
e.g. If n  then  e.g. If n  then  2
3 3 3 3 6 6 6
 n
m  ax
Type I: Examples of type x e dx then put ax n  t
0
Evaluate the following integrals:

n 1  kx
1)  x e dx
0

Answer: Let I   x n 1e  kxdx
0
t dt
Put k x  t x  dx  x 0 
k k t 0 
n 1

t  t  dt 1   t n 1
n 
I  e   I  e t dt
0 k k k 0

n
I 
kn

h2 x 2
2)  e dx
0
2 2 
Answer: Let I   e  h x dx
0
1
t 
 dx  t  dt
1 x 0
Put h 2 x 2  t  hx  t 2 x
h 2h t 0 



Faculty of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur Page 2

, 
 I   e t
1 
t dt
0
2h

1   t   1 1
I   e t dt
2h 0
 n  1    n  
 2 2

1
2 
I  I 
2h 2h
 2
7 2 x
3) x e dx
0
 2
Answer: Let I   x 7 e  2 x dx
0

Put 2 x 2  t  2 x t

t
t  dt
1
x  dx  x 0 
21/ 2 1/ 2
2* 2 t 0 

 7
 t 
 I   e t  t  dt
 1
  2*
0  

1  t 3
n  1  3  n  4
5 
I  e t dt
2 0

4 3! 6 3
I  I  I  
25 32 32 16
 3
4)  x e  x dx
0
 3
Answer: Let I   x e  x dx
0

Put x  t  x  t3  dx  3t 2 dt x 0 
t 0 


 1/ 2 2
 I   e t t 3 3t dt
0


Faculty of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur Page 3

, 
 9
 I  3  e  t t  dt
7
 n  1   n  
0  2 2

9 9 7 7 75 5 753 3 7531 1
I  3     
2 2 2 2 22 2 222 2 2222 2

7531 1
I  3
2222 2

315
I  
16
 
 x8 5 x 8
5)  xe dx  x e dx
0 0
 
 x8 8
5 x
Answer: Let I   xe dx  x e dx
0 0
 I  I1  I 2 ......(1 )
 8
 I1   xe  x dx
0

Put x8  t  x  t

 dx  t  dt
1 X 0 
8 T 0 

 I1   e  t t t  dt
1
0
8

1  t   3 1
 I1   e t dt  n  1    n  
80  4 4

1 1
 I1 
8 4
 8
5 x
and I 2   x e dx
0


 
 5 1 
 I 2   e  t t t dt
0
8


Faculty of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur Page 4