Module -1

Partial differentiation

Question and answer

1. Evaluate   0∞  x3/2  e -x dx

Solution:               0∞  x3/2  e -x dx          =   0∞  x 5/2-1  e -x dx  

=  γ(5/2)

         =   γ(3/2+ 1) 

         =    3/2   γ(3/2 )  

        =   3/2 .  ½   γ(½ ) 

=  3/2 . ½ .π 

        =    ¾ π  

2. Find  γ(-½) 

Solution:    (-½) + 1  =  ½
γ(-1/2)  =   γ(-½ + 1)  / (-½)   

=   - 2   γ(1/2 )  

=  - 2 π  

3. Show that    

Solution : =

                                          =

                                          = )  .......................

                                          =

                                          =

4.  Evaluate  dx.

Solution :  Let    dx 

X	0
t	0

                                                    Put or ;dx =2t dt .

dt

dt

Ex. 5: Evaluate   dx.

Solution :   Let   dx.

x	0
t	0

                                        Put or  ;             4x dx = dt

dx

6. Evaluate:  I =  02  x2  / (2 – x )   . Dx

Solution:

Letting   x  = 2y, we get

I   =   (8/2) 01  y 2  (1 – y ) -1/2dy

=  (8/2) . B(3 , 1/2 )

= 642 /15

7. Evaluate

Solution :Let

Put   or  ,,

When,;,

	o
	1	0

Also

8. Prove that

Solution : Let

Put  or  , 

9. Show that

Solution :

                          =

(0<p<1)

(by above result)  

10. Evaluate:  I =  02  x2  / (2 – x )   . Dx

Solution:

Letting   x  = 2y, we get

I   =   (8/2) 01  y 2  (1 – y ) -1/2dy

=  (8/2) . B(3 , 1/2 )

= 642 /15