JEE Mathematics/ Lakshmikanta Satapathy/ Questions on Indefinite integration solved by rationalization and substitution, involving polynomial and trigonometric functions

1. Physics Helpline
L K Satapathy
Indefinite Integrals 7

2. Physics Helpline
L K Satapathy
Indefinite Integrals - 7
Answer-1
  
.
dx x a x b
I dx
x a x b x a x b x a x b
  
 
        
 
. .
( ) ( )
x a x b x a x b
dx dx
x a x b a b
     
 
    
 1
. .x a dx x b dx C
a b
    
  
3 2 3 2
1 ( ) ( )
3 2 3 2
x a x b
C
a b
  
   
  
3 2 3 22
( [ ]) ( )
3( )
x a x b C
a
ns
b
A      

3. Physics Helpline
L K Satapathy
Indefinite Integrals - 7
Answer-2 2 2 2 2
sin2
.
sin cos
x
I dx
a x b x


Substitution : 2 2 2 2
sin cosPut a x b x t 
2 2
[ .2sin cos .2cos ( sin )].a x x b x x dx dt   
2 2
[ .sin2 .sin2 ].a x b x dx dt  
2 2
2 2
( )sin2 . sin2 .
dt
a b x dx dt x dx
a b
    

2 2 2 2
1 1
log
dt
I t C
a b t a b
   
 
 
2 2 2 2
2 2
1
log sin [c ]osa x b x C
a
Ans
b
  


4. Physics Helpline
L K Satapathy
Indefinite Integrals - 7
Answer-3
sin sin[( ) ]
. .
sin( ) sin( )
x x a a
I dx dx
x a x a
 
 
  
sin( )cos cos( )sin
.
sin( )
x a a x a a
dx
x a
  


cos( )
cos sin .
sin( )
x a
a dx a dx
x a

 
 
cos sin .
dt
I a dx a dx
t
   
sin( ) cos( ).Put x a t x a dx dt    
(cos ) sin loga x a t C  
(cos ) (sin )log s [n ]i ( )a x a x Ansa C   
Substitution :

5. Physics Helpline
L K Satapathy
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