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NCERT Solutions Class 12 Maths Chapter 7 - Integrals

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1 	 / 	 2 9
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
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
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
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
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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FAQs on NCERT Solutions Class 12 Maths Chapter 7 - Integrals

1. What are integrals?
Ans. Integrals are mathematical tools used to calculate the area under a curve or to find the accumulation of quantities over a given interval. They are fundamental in calculus and have various applications in physics, engineering, economics, and other fields.
2. How do you find the integral of a function?
Ans. To find the integral of a function, you can use integration techniques such as the power rule, substitution, integration by parts, or trigonometric substitutions. These techniques allow you to manipulate the function and evaluate the integral using specific formulas or rules.
3. What is the difference between definite and indefinite integrals?
Ans. A definite integral has upper and lower limits that define a specific interval over which the integration is performed. It gives a numerical value representing the area or accumulation within that interval. On the other hand, an indefinite integral does not have any limits and represents a family of functions obtained by adding a constant (known as the constant of integration) to the result.
4. Can integrals be used to solve real-life problems?
Ans. Yes, integrals are widely used to solve real-life problems. For example, they can be used to calculate the area under a curve, which can represent the total distance traveled, the total amount of a substance dissolved, or the total revenue generated over a given time period. Integrals are also used in physics to calculate quantities such as work, fluid flow, or electric charge.
5. What are the applications of integrals in engineering?
Ans. Integrals have various applications in engineering. They can be used to determine the moment of inertia of a solid object, calculate the displacement and velocity of a particle, find the center of mass of a system, analyze electrical circuits, and solve differential equations that model physical systems. These applications help engineers in designing and optimizing structures, systems, and processes.
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