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NCERT Solutions Class 12 Maths Chapter 7 - Integrals

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1 	 / 	 3 1
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 . 4
I n t e g r a t e 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
I 	 = 	 	
= 	 	 	 	 	 	 	 	 	 	 	 	 u s i n g 	
= 	 	 	 A n s .
2 . 	 	
Page 2


1 	 / 	 3 1
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 . 4
I n t e g r a t e 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
I 	 = 	 	
= 	 	 	 	 	 	 	 	 	 	 	 	 u s i n g 	
= 	 	 	 A n s .
2 . 	 	
2 	 / 	 3 1
A n s . 	
= 	
= 	
	
= 	
3 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
Page 3


1 	 / 	 3 1
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 . 4
I n t e g r a t e 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
I 	 = 	 	
= 	 	 	 	 	 	 	 	 	 	 	 	 u s i n g 	
= 	 	 	 A n s .
2 . 	 	
2 	 / 	 3 1
A n s . 	
= 	
= 	
	
= 	
3 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
3 	 / 	 3 1
4 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
5 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
Page 4


1 	 / 	 3 1
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 . 4
I n t e g r a t e 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
I 	 = 	 	
= 	 	 	 	 	 	 	 	 	 	 	 	 u s i n g 	
= 	 	 	 A n s .
2 . 	 	
2 	 / 	 3 1
A n s . 	
= 	
= 	
	
= 	
3 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
3 	 / 	 3 1
4 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
5 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
4 	 / 	 3 1
I 	 = 	 	
= 	 	
= 	 	
= 	 	
= 	 	 A n s .
6 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	
= 	 	 … … … . ( i )
P u t t i n g 	
	
	 	
F r o m 	 e q . 	 ( i ) ,
Page 5


1 	 / 	 3 1
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 . 4
I n t e g r a t e 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
I 	 = 	 	
= 	 	 	 	 	 	 	 	 	 	 	 	 u s i n g 	
= 	 	 	 A n s .
2 . 	 	
2 	 / 	 3 1
A n s . 	
= 	
= 	
	
= 	
3 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
3 	 / 	 3 1
4 . 	 	
A n s . 	 	
= 	 	
= 	 	
= 	 	
5 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	 	 … … … . ( i )
P u t t i n g 	
	 	
	 	
	 	 F r o m 	 e q . 	 ( i ) ,
4 	 / 	 3 1
I 	 = 	 	
= 	 	
= 	 	
= 	 	
= 	 	 A n s .
6 . 	 	
A n s . 	 L e t 	 I 	 = 	
= 	
= 	 	 … … … . ( i )
P u t t i n g 	
	
	 	
F r o m 	 e q . 	 ( i ) ,
5 	 / 	 3 1
I 	 = 	
= 	 	
= 	
	
	 = 	 	
7 . 	 	
A n s . 	 L e t 	 I 	 = 	 	
= 	 	
= 	 	
L e t 	 I
1
	 = 	 	
P u t t i n g 	
	
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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 total accumulated quantity or area under a curve. They are a fundamental concept in calculus and help in solving problems related to finding areas, volumes, and even averages.
2. How are integrals helpful in real-life applications?
Ans. Integrals have various real-life applications. For example, they can be used to calculate the total distance traveled by a moving object when its velocity is known. They are also used in physics to find the work done by a force, calculate the center of mass of an object, and determine the amount of fluid flowing through a pipe.
3. What is the difference between indefinite and definite integrals?
Ans. Indefinite integrals and definite integrals are different in terms of their outcomes. An indefinite integral gives a family of functions as its result, while a definite integral provides a single numerical value. In other words, an indefinite integral represents an antiderivative, whereas a definite integral gives the net accumulation or total value of a function over a specific interval.
4. How can integrals be evaluated?
Ans. Integrals can be evaluated using various methods such as basic integration rules, substitution, integration by parts, and partial fractions. The choice of method depends on the complexity of the integrand and the available techniques. It is important to practice different integration techniques to become proficient in evaluating integrals.
5. Can integrals be used to solve optimization problems?
Ans. Yes, integrals can be used to solve optimization problems. By formulating the problem as a mathematical function and finding the maximum or minimum value of that function using calculus techniques, integrals can help determine the optimal solution. This is commonly done in economics, engineering, and other fields where maximizing or minimizing a certain quantity is desired.
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