NCERT Solutions - Integrals, Exercise - 7.5 JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : NCERT Solutions - Integrals, Exercise - 7.5 JEE Notes | EduRev

 Page 1


1 	 / 	 2 9
C B S E 	 C l a s s – 1 0 	 M a t h e m a t i c s
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 5
I n t e g r a t e 	 t h e 	 ( r a t i o n a l ) 	 f u n c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 	 . . . . . . . ( i )
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 	 o n 	 b o t h 	 s i d e s 	 A 	 + 	 B 	 = 	 1 	 . . . . . ( i i )
C o m p a r i n g 	 c o n s t a n t s 	 2 A 	 + 	 B 	 = 	 0 	 . . . . . ( i i i )
S o l v i n g 	 e q . 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 a n d 	 B 	 = 	 2
P u t t i n g 	 t h e s e 	 v a l u e s 	 o f 	 A 	 a n d 	 B 	 i n 	 e q . 	 ( i ) ,
	
= 	
Page 2


1 	 / 	 2 9
C B S E 	 C l a s s – 1 0 	 M a t h e m a t i c s
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 5
I n t e g r a t e 	 t h e 	 ( r a t i o n a l ) 	 f u n c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 	 . . . . . . . ( i )
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 	 o n 	 b o t h 	 s i d e s 	 A 	 + 	 B 	 = 	 1 	 . . . . . ( i i )
C o m p a r i n g 	 c o n s t a n t s 	 2 A 	 + 	 B 	 = 	 0 	 . . . . . ( i i i )
S o l v i n g 	 e q . 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 a n d 	 B 	 = 	 2
P u t t i n g 	 t h e s e 	 v a l u e s 	 o f 	 A 	 a n d 	 B 	 i n 	 e q . 	 ( i ) ,
	
= 	
2 	 / 	 2 9
= 	
2 . 	
A n s . 	
= 	
= 	
= 	
3 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 A 	 + 	 B 	 + 	 C 	 = 	 0 	 . . . . . . . ( i i )
Page 3


1 	 / 	 2 9
C B S E 	 C l a s s – 1 0 	 M a t h e m a t i c s
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 5
I n t e g r a t e 	 t h e 	 ( r a t i o n a l ) 	 f u n c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 	 . . . . . . . ( i )
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 	 o n 	 b o t h 	 s i d e s 	 A 	 + 	 B 	 = 	 1 	 . . . . . ( i i )
C o m p a r i n g 	 c o n s t a n t s 	 2 A 	 + 	 B 	 = 	 0 	 . . . . . ( i i i )
S o l v i n g 	 e q . 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 a n d 	 B 	 = 	 2
P u t t i n g 	 t h e s e 	 v a l u e s 	 o f 	 A 	 a n d 	 B 	 i n 	 e q . 	 ( i ) ,
	
= 	
2 	 / 	 2 9
= 	
2 . 	
A n s . 	
= 	
= 	
= 	
3 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 A 	 + 	 B 	 + 	 C 	 = 	 0 	 . . . . . . . ( i i )
3 	 / 	 2 9
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 – 5 A 	 – 	 4 B 	 – 	 3 C = 	 3
	 5 A 	 + 	 4 B 	 + 	 3 C 	 = 	 – 3 	 . . . . . . . ( i i i )
C o m p a r i n g 	 c o n s t a n t s : 	 6 A 	 + 	 3 B 	 + 	 2 C 	 = 	 – 1 	 . . . . . . . ( i v )
O n 	 s o l v i n g 	 e q . 	 ( i ) , 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 1 , 	 B 	 = 	 – 5 , 	 C 	 = 	 4
P u t t i n g 	 t h e 	 v a l u e s 	 o f 	 A , 	 B 	 a n d 	 C 	 i n 	 e q . 	 ( i ) ,
= 	
	
= 	
	
= 	
4 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
Page 4


1 	 / 	 2 9
C B S E 	 C l a s s – 1 0 	 M a t h e m a t i c s
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 5
I n t e g r a t e 	 t h e 	 ( r a t i o n a l ) 	 f u n c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 	 . . . . . . . ( i )
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 	 o n 	 b o t h 	 s i d e s 	 A 	 + 	 B 	 = 	 1 	 . . . . . ( i i )
C o m p a r i n g 	 c o n s t a n t s 	 2 A 	 + 	 B 	 = 	 0 	 . . . . . ( i i i )
S o l v i n g 	 e q . 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 a n d 	 B 	 = 	 2
P u t t i n g 	 t h e s e 	 v a l u e s 	 o f 	 A 	 a n d 	 B 	 i n 	 e q . 	 ( i ) ,
	
= 	
2 	 / 	 2 9
= 	
2 . 	
A n s . 	
= 	
= 	
= 	
3 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 A 	 + 	 B 	 + 	 C 	 = 	 0 	 . . . . . . . ( i i )
3 	 / 	 2 9
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 – 5 A 	 – 	 4 B 	 – 	 3 C = 	 3
	 5 A 	 + 	 4 B 	 + 	 3 C 	 = 	 – 3 	 . . . . . . . ( i i i )
C o m p a r i n g 	 c o n s t a n t s : 	 6 A 	 + 	 3 B 	 + 	 2 C 	 = 	 – 1 	 . . . . . . . ( i v )
O n 	 s o l v i n g 	 e q . 	 ( i ) , 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 1 , 	 B 	 = 	 – 5 , 	 C 	 = 	 4
P u t t i n g 	 t h e 	 v a l u e s 	 o f 	 A , 	 B 	 a n d 	 C 	 i n 	 e q . 	 ( i ) ,
= 	
	
= 	
	
= 	
4 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
4 	 / 	 2 9
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 A 	 + 	 B 	 + 	 C 	 = 	 0 	 . . . . . . . ( i i )
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 – 5 A 	 – 	 4 B 	 – 	 3 C = 	 1
	 5 A 	 + 	 4 B 	 + 	 3 C 	 = 	 – 1 	 . . . . . . . ( i i i )
C o m p a r i n g 	 c o n s t a n t s : 	 6 A 	 + 	 3 B 	 + 	 2 C 	 = 	 0 	 . . . . . . . ( i v )
O n 	 s o l v i n g 	 e q . 	 ( i ) , 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 B 	 = 	 – 2 , 	 C 	 = 	
P u t t i n g 	 t h e 	 v a l u e s 	 o f 	 A , 	 B 	 a n d 	 C 	 i n 	 e q . 	 ( i ) ,
= 	
	
= 	
	
= 	
5 . 	
Page 5


1 	 / 	 2 9
C B S E 	 C l a s s – 1 0 	 M a t h e m a t i c s
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 5
I n t e g r a t e 	 t h e 	 ( r a t i o n a l ) 	 f u n c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 	 . . . . . . . ( i )
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 	 o n 	 b o t h 	 s i d e s 	 A 	 + 	 B 	 = 	 1 	 . . . . . ( i i )
C o m p a r i n g 	 c o n s t a n t s 	 2 A 	 + 	 B 	 = 	 0 	 . . . . . ( i i i )
S o l v i n g 	 e q . 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 a n d 	 B 	 = 	 2
P u t t i n g 	 t h e s e 	 v a l u e s 	 o f 	 A 	 a n d 	 B 	 i n 	 e q . 	 ( i ) ,
	
= 	
2 	 / 	 2 9
= 	
2 . 	
A n s . 	
= 	
= 	
= 	
3 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 A 	 + 	 B 	 + 	 C 	 = 	 0 	 . . . . . . . ( i i )
3 	 / 	 2 9
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 – 5 A 	 – 	 4 B 	 – 	 3 C = 	 3
	 5 A 	 + 	 4 B 	 + 	 3 C 	 = 	 – 3 	 . . . . . . . ( i i i )
C o m p a r i n g 	 c o n s t a n t s : 	 6 A 	 + 	 3 B 	 + 	 2 C 	 = 	 – 1 	 . . . . . . . ( i v )
O n 	 s o l v i n g 	 e q . 	 ( i ) , 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 1 , 	 B 	 = 	 – 5 , 	 C 	 = 	 4
P u t t i n g 	 t h e 	 v a l u e s 	 o f 	 A , 	 B 	 a n d 	 C 	 i n 	 e q . 	 ( i ) ,
= 	
	
= 	
	
= 	
4 . 	
A n s . 	
= 	 	 . . . . . . . ( i )
	
4 	 / 	 2 9
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 A 	 + 	 B 	 + 	 C 	 = 	 0 	 . . . . . . . ( i i )
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 : 	 – 5 A 	 – 	 4 B 	 – 	 3 C = 	 1
	 5 A 	 + 	 4 B 	 + 	 3 C 	 = 	 – 1 	 . . . . . . . ( i i i )
C o m p a r i n g 	 c o n s t a n t s : 	 6 A 	 + 	 3 B 	 + 	 2 C 	 = 	 0 	 . . . . . . . ( i v )
O n 	 s o l v i n g 	 e q . 	 ( i ) , 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 B 	 = 	 – 2 , 	 C 	 = 	
P u t t i n g 	 t h e 	 v a l u e s 	 o f 	 A , 	 B 	 a n d 	 C 	 i n 	 e q . 	 ( i ) ,
= 	
	
= 	
	
= 	
5 . 	
5 	 / 	 2 9
A n s . 	
= 	
= 	
= 	
= 	 	 . . . . ( i )
	
	
C o m p a r i n g 	 c o e f f i c i e n t s 	 o f 	 	 o n 	 b o t h 	 s i d e s 	 A 	 + 	 B 	 = 	 2 	 . . . . . . . ( i i )
C o m p a r i n g 	 c o n s t a n t s 	 2 A 	 + 	 B 	 = 	 0 	 . . . . . ( i i i )
S o l v i n g 	 e q . 	 ( i i ) 	 a n d 	 ( i i i ) , 	 w e 	 g e t 	 A 	 = 	 	 a n d 	 B 	 = 	 4
P u t t i n g 	 t h e s e 	 v a l u e s 	 o f 	 A 	 a n d 	 B 	 i n 	 e q . 	 ( i ) ,
	
	
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