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NCERT Solutions Class 12 Maths Chapter 7 - Integrals

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1 	 / 	 2 5
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 . 0 1
B y 	 u s i n g 	 t h e 	 p r o p e r t i e s 	 o f 	 d e f i n i t e 	 i n t e g r a l s , 	 e v a l u a t e 	 t h e 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . ( i )
= 	
	
	 	 	 	 	 	 	 	 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
Page 2


1 	 / 	 2 5
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 . 0 1
B y 	 u s i n g 	 t h e 	 p r o p e r t i e s 	 o f 	 d e f i n i t e 	 i n t e g r a l s , 	 e v a l u a t e 	 t h e 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . ( i )
= 	
	
	 	 	 	 	 	 	 	 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
2 	 / 	 2 5
	 2 I 	 = 	
	 I 	 = 	 	 A n s .
2 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
Page 3


1 	 / 	 2 5
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 . 0 1
B y 	 u s i n g 	 t h e 	 p r o p e r t i e s 	 o f 	 d e f i n i t e 	 i n t e g r a l s , 	 e v a l u a t e 	 t h e 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . ( i )
= 	
	
	 	 	 	 	 	 	 	 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
2 	 / 	 2 5
	 2 I 	 = 	
	 I 	 = 	 	 A n s .
2 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
3 	 / 	 2 5
= 	
	 2 I 	 = 	
	
	 2 I 	 = 	
	 I 	 = 	 	 A n s . 	
3 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 	 … … … . ( i i )
Page 4


1 	 / 	 2 5
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 . 0 1
B y 	 u s i n g 	 t h e 	 p r o p e r t i e s 	 o f 	 d e f i n i t e 	 i n t e g r a l s , 	 e v a l u a t e 	 t h e 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . ( i )
= 	
	
	 	 	 	 	 	 	 	 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
2 	 / 	 2 5
	 2 I 	 = 	
	 I 	 = 	 	 A n s .
2 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
3 	 / 	 2 5
= 	
	 2 I 	 = 	
	
	 2 I 	 = 	
	 I 	 = 	 	 A n s . 	
3 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 	 … … … . ( i i )
4 	 / 	 2 5
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
	 2 I 	 = 	 	
	 2 I 	 = 	
	 I 	 = 	 	 A n s .
4 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . . ( i )
	 I 	 = 	
Page 5


1 	 / 	 2 5
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 . 0 1
B y 	 u s i n g 	 t h e 	 p r o p e r t i e s 	 o f 	 d e f i n i t e 	 i n t e g r a l s , 	 e v a l u a t e 	 t h e 	 i n t e g r a l s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 6 .
1 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . ( i )
= 	
	
	 	 	 	 	 	 	 	 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
2 	 / 	 2 5
	 2 I 	 = 	
	 I 	 = 	 	 A n s .
2 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
3 	 / 	 2 5
= 	
	 2 I 	 = 	
	
	 2 I 	 = 	
	 I 	 = 	 	 A n s . 	
3 . 	
A n s . 	 L e t 	 I 	 = 	 	 	 … … … . . ( i )
	 I 	 = 	
	 I 	 = 	 	 	 … … … . ( i i )
4 	 / 	 2 5
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
	 2 I 	 = 	 	
	 2 I 	 = 	
	 I 	 = 	 	 A n s .
4 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . . ( i )
	 I 	 = 	
5 	 / 	 2 5
	 I 	 = 	 	 … … … . ( i i )
A d d i n g 	 e q . 	 ( i ) 	 a n d 	 ( i i ) ,
2 I 	 = 	
= 	
	 2 I 	 = 	 	
	 2 I 	 = 	 	 	
	 I 	 = 	 	 A n s .
5 . 	
A n s . 	 L e t 	 I 	 = 	 	 … … … . ( i )
P u t t i n g 	
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FAQs on NCERT Solutions Class 12 Maths Chapter 7 - Integrals

1. What is an integral?
Ans. An integral is a mathematical concept used to find the area under a curve or the accumulation of a quantity over an interval. It is denoted by the symbol ∫ and is a fundamental operation in calculus.
2. How do you find the integral of a function?
Ans. To find the integral of a function, you need to apply integration rules and techniques. These include power rule, substitution, integration by parts, and partial fractions. By using these methods, you can evaluate the integral and obtain a solution.
3. What is the significance of integrals in real-life applications?
Ans. Integrals have various real-life applications, such as calculating areas of irregular shapes, finding the total distance traveled by an object with varying velocity, determining the amount of accumulated quantities like liquid or population growth, and analyzing continuous data in fields like physics, economics, and engineering.
4. Are there different types of integrals?
Ans. Yes, there are different types of integrals. The most common type is the definite integral, which calculates the area under a curve between two specified limits. Another type is the indefinite integral, which finds the antiderivative of a function. There are also improper integrals, which deal with functions that have infinite limits or discontinuities.
5. How can I practice solving integrals?
Ans. To practice solving integrals, you can start by using textbooks or online resources that provide a variety of integral problems with solutions. Additionally, you can attempt past exam papers or seek out practice worksheets specifically designed for integral calculus. Working through these problems and seeking guidance when needed will help improve your skills in solving integrals.
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