The document provides solutions to indefinite integrals presented by L.K. Satapathy, detailing various substitution methods and step-by-step integrative processes. It covers different types of integrals, including those involving trigonometric functions and algebraic terms. Additional information about the physics helpline and its resources is also included.

1. Physics Helpline
L K Satapathy
Indefinite Integrals 12

2. Physics Helpline
L K Satapathy
Indefinite Integrals - 12
Answer-1 ( )( )
. .
( )( )
a x a x a x
I dx dx
a x a x a x
  
 
   
2 2
2 , 2 . .
2
dt
For I Put a x t x dx dt x dx      Substitution :
1 22 2 2 2 2 2
.
. .
a x a x dx
dx dx I I
a x a x a x
 
    
  
  
1
1 2 2
sin
dx x
I a a
aa x

  

 2 2 2
.x dx
and I
a x




2 2
2
2
dt
I t C a x C
t
      
1 2 2
1 2 [s ]in
x
I I I a a AnC sx
a

      

3. Physics Helpline
L K Satapathy
Indefinite Integrals - 12
Answer-2
Substitution : 2 2
tan sec . sec .
dt
Put a x t a x dx dt x dx
a
    
2
2 2 2 2 2 2 2
sec .
sin cos tan
dx x dx
I
a x b x a x b
 
  
2 2
1 dt
I
a t b


11 1
tan
t
C
a b b
 
  
 
1
[ ]
1 tan
tan
a x
C
ab b
Ans  
  
 
11
tan
t
C
ab b

 
1
2 2
1
tan
dx x
using C
x a a a
  
    


4. Physics Helpline
L K Satapathy
Indefinite Integrals - 12
Answer-3
2
16 6
dx
I
x x

 

2
25 ( 6 9)
dx
x x

  

2
25 9 6
dx
x x

  

2 2
5 ( 3)
dx
x

 

1
[
3
sin
5
]
x
C Ans  
  
 
1
2 2
sin
dx x
using C
aa x
  
  
 


5. Physics Helpline
L K Satapathy
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