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NCERT Solutions Class 12 Maths Chapter 7 - Integrals

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1 	 / 	 1 3
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 . 1
F i n d 	 a n 	 a n t i d e r i v a t i v e 	 ( o r 	 i n t e g r a l ) 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 b y 	 t h e 	 m e t h o d 	 o f
i n s p e c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 5 .
1 . 	 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
2 . 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
Page 2


1 	 / 	 1 3
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 . 1
F i n d 	 a n 	 a n t i d e r i v a t i v e 	 ( o r 	 i n t e g r a l ) 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 b y 	 t h e 	 m e t h o d 	 o f
i n s p e c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 5 .
1 . 	 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
2 . 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
3 . 	 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
4 . 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
5 . 	 	
A n s . 	
Page 3


1 	 / 	 1 3
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 . 1
F i n d 	 a n 	 a n t i d e r i v a t i v e 	 ( o r 	 i n t e g r a l ) 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 b y 	 t h e 	 m e t h o d 	 o f
i n s p e c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 5 .
1 . 	 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
2 . 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
3 . 	 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
4 . 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
5 . 	 	
A n s . 	
	 	 … . . ( i )
A g a i n 	 	
	 	
	 	 [ M u l t i p l y i n g 	 b o t h 	 s i d e s 	 b y 	 ] 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) , 	 w e 	 g e t
	 	
	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
E v a l u a t e 	 t h e 	 f o l l o w i n g 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 6 	 t o 	 1 1 .
6 . 	 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	 	
Page 4


1 	 / 	 1 3
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 . 1
F i n d 	 a n 	 a n t i d e r i v a t i v e 	 ( o r 	 i n t e g r a l ) 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 b y 	 t h e 	 m e t h o d 	 o f
i n s p e c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 5 .
1 . 	 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
2 . 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
3 . 	 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
4 . 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
5 . 	 	
A n s . 	
	 	 … . . ( i )
A g a i n 	 	
	 	
	 	 [ M u l t i p l y i n g 	 b o t h 	 s i d e s 	 b y 	 ] 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) , 	 w e 	 g e t
	 	
	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
E v a l u a t e 	 t h e 	 f o l l o w i n g 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 6 	 t o 	 1 1 .
6 . 	 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	 	
7 . 	
A n s . 	 	 = 	 	 = 	 	 = 	 	
	 	
8 . 	 	
A n s . 	 	
= 	 	
= 	
= 	 	 w h e r e 	 	 i s 	 t h e 	 c o n s t a n t 	 o f 	 i n t e g r a t i o n .
9 . 	 	
A n s . 	 	
= 	 	
= 	
= 	 	
Page 5


1 	 / 	 1 3
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 . 1
F i n d 	 a n 	 a n t i d e r i v a t i v e 	 ( o r 	 i n t e g r a l ) 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 b y 	 t h e 	 m e t h o d 	 o f
i n s p e c t i o n 	 i n 	 E x e r c i s e s 	 1 	 t o 	 5 .
1 . 	 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
2 . 	
A n s . 	
	
	 	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
3 . 	 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
4 . 	
A n s . 	 	 	
	
	 	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
5 . 	 	
A n s . 	
	 	 … . . ( i )
A g a i n 	 	
	 	
	 	 [ M u l t i p l y i n g 	 b o t h 	 s i d e s 	 b y 	 ] 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) , 	 w e 	 g e t
	 	
	
	 A n 	 a n t i - d e r i v a t i v e 	 o f 	 	 i s 	 	
E v a l u a t e 	 t h e 	 f o l l o w i n g 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 6 	 t o 	 1 1 .
6 . 	 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	 	
7 . 	
A n s . 	 	 = 	 	 = 	 	 = 	 	
	 	
8 . 	 	
A n s . 	 	
= 	 	
= 	
= 	 	 w h e r e 	 	 i s 	 t h e 	 c o n s t a n t 	 o f 	 i n t e g r a t i o n .
9 . 	 	
A n s . 	 	
= 	 	
= 	
= 	 	
1 0 . 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
= 	 	
1 1 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
= 	 	
= 	 	
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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. They help in finding the total accumulation of a quantity over a given interval.
2. How are integrals useful in real-life applications?
Ans. Integrals are widely used in various fields such as physics, economics, engineering, and statistics. They can be used to determine the total distance traveled, calculate the area of irregular shapes, find the average value of a function, and solve problems related to rates of change.
3. What is the process of finding an integral?
Ans. The process of finding an integral involves taking the antiderivative of a function. It is represented by the symbol ∫ and is followed by the function to be integrated and the variable of integration. The result is a new function, known as the antiderivative or the indefinite integral.
4. What are the different types of integrals?
Ans. There are two main types of integrals: definite and indefinite integrals. Definite integrals have upper and lower limits, and they give a specific value as the result. Indefinite integrals do not have limits and represent a family of functions.
5. How can I solve integrals?
Ans. Integrals can be solved using various techniques such as substitution, integration by parts, partial fractions, trigonometric identities, and special rules like the power rule or the chain rule. It is important to understand the properties and rules of integration to effectively solve integrals.
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