The document discusses the evaluation of various definite integrals, specifically involving trigonometric and exponential functions. It includes detailed calculations and transformations applied to integrals to derive results. Additional resources and platforms for further assistance in physics are also provided.

1. Physics Helpline
L K Satapathy
Definite Integrals 8

2. Physics Helpline
L K Satapathy
Definite Integrals - 8
Question : The value of is equal to
22
2
cos
1 x
x x
dx
e




2 2
2 22 2
( ) 2 ( ) 2 ( ) ( )
4 4
a b c e d e
 
 
    
Answer :
22
2
cos
1 x
x x
I dx
e





[Putting x = – t ]
22
2
cos
( )
1 t
t t
dt
e




 

The given integral
22
2
cos
1 t
t t
dt
e






. . . (1)
( ) ( )
b a
a b
f x dx f x dx
 
  
 
 

3. Physics Helpline
L K Satapathy
Definite Integrals - 8
22
2
cos
1 x
x x
I dx
e




 

2
2
2
cosx xdx



 
22
2
cos
1
x
x
x e x
I dx
e



 

[Replacing t by x ]
[ Multiplying numerator
& denominator by ]x
e
. . . (2)
[ Adding equations (1) & (2) ]
22
2
cos (1 )
2
1
x
x
x x e
I dx
e




 


4. Physics Helpline
L K Satapathy
Definite Integrals - 8
2
22
0
0
sin 2 sinI x x x xdx


     
2
2
0
2 2 cosI x xdx

   [ Since is an even function]
2
2
0
cosI x xdx

  
2
cosx x
Integrating by parts :
2
& cosx u xdx dv  2 & sindu xdx v x  
udv uv vdu  
22
0
2 sin
4
I x xdx


    . . . (3)

5. Physics Helpline
L K Satapathy
Definite Integrals - 8
Correct option = (a)
Again integrating by parts :sinx xdx
& sin & cosx u xdx dv du dx v x     
sin ( cos ) cosx xdx x x xdx     
cos cosx x xdx   
cos sinx x x  
   
2
2 2
0 0
0
sin cos sin 0 1 1x xdx x x x

 
      
22 2
0
(3) 2 sin 2
4 4
[ ]I x xd sx An

 
    

6. Physics Helpline
L K Satapathy
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