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Math Assignment Class XII Ch -07 | Integrals

 Math Assignment  Class XII  Ch -07 | Integrals

Extra questions of chapter 07 Integrals class XII  with answers and  hints to the difficult questions, strictly according to the CBSE Board syllabus. Important and useful math. 

MATHEMATICS ASSIGNMENT OF EXTRA QUESTION

STRICTLY ACCORDING TO THE PREVIOUS CBSE BOARD SAMPLE QUESTION PAPERS
FROM 2018 TO 2022

Find the following integrals

Question 1

equation

Answer:  4

Question 2

equation

Answer  equation

Question 3

equation

Answer     equation

Question 4

equation

Answer    equation

Question 5
 equation
Answer:  0
Solution Hint :  x2sinx is an odd function so this integral equal to zero

Question 6

equation

Answer:  I = ex(1 - cot x) + C

Solution Hint

f(x) = 1 - cot x   ⇒  f ' (x) = cosec2

equation

I = ex(1 - cot x)+C

Question 7


equation

Answer    equation

Question 8

equation

Answer    equation

Solution Hint

equation

Now putting 1 - tan x = t

Question 9

equation

Answer   equation

Solution Hint

equation

Question 10

Evaluate:  equation 

Answer     equation

Solution Hint
Taking '-' common from the denominator and then applying method of completing the square.

Question 11

Evaluate:  equation

Answer:  π / 12

Solution Hint:

equation   ......(1) 

equation

equation ..... (2)

Adding equation (1) and equation (2) we get

equation

Question 12

Evaluate:  equation

Answer :  5

Solution Hint

x - 1 = 0 when x = 1, so given integral becomes

equation

Integrate and putting the limit we get  I = 5

Question 13

equation

Answer:  2

Question 14

equation

Answer   equation

Solution Hint

equation 

equation 
equation

Now integrating the first integral by parts we get

equation

equation

equation

equation

Question 15

equation ...... (1)

Answer   equation

Solution Hint

Putting cos2x = t   ⇒  - 2 cosx sinx dx = dt  ⇒  Sin2x dx = - dt

Putting these values in equation (1) we get

equation

equation

equation

Question 16

equation

Answer :  I = 11/4     

Solution Hint:  Factorise the given function we get 

equation

x(x - 1)(x - 2) = 0 ⇒ x = 0, 1, 2

Given integral can be written as 

equation

equation

Find the integrals and putting the limits we get

equation

Question 17

equation

Answer   equation

Solution Hint:

Putting x2 = t then use partial fraction  and then integrate

Question 18

equation

Answer:  tan x - tan-1x + C

Solution Hint:

Adding and subtracting '1' in the numerator we get


equation

Separating the denominator we get

equation

equation

I = tan x - tan-1x + C

Question 19

equation

Answer   equation

Solution Hint

equation

Separating the denominator we get

equation

equation

Question 20

equation

Answer  equation

Solution Hint

Taking x4 common from the numerator we get

equation

equation ............. (1)

equation 

equation

Differentiating both side we get

equation

Putting all these values in eqn. (1) we get

equation

equation

Question 21
equation 
Answer: 2log2

Solution Hint

equation

I = I1 + I2 

Since Iis an odd function so I1 = 0
equation

Since is an even function so 
equation

equation

equation

⇒ I = 2log2
Question 22
equation

Answer   equation
Solution Hint:

Multiply and divide  by x we get

equation

Now putting x2 = t   ⇒  2xdx = dt   ⇒ xdx = dt/2  we get

equation

Using partial fraction to solve this integral

equation

Solving these fractions we get,  A = 1, B = 0, C = - 2 and the given integral becomes
equation

equation

equation

equation

equation

Question 23
equation

Answer:   equation
Solution Hint

equation

equation

equation 

equation

equation
equation
equation
Question 24
equation

Answer

equation
Solution Hint

equation

equation

equation 

equation 

equation

Putting sin x = t  ⇒ cosx dx = dt  we get

equation

Now using partial fraction we get

equation

equation
Now putting  t = sinx we get

equation
Question 25

equation 

Answer   equation
Solution Hint

equation

equation  

equation 

⇒ f(x) = -f(x) ⇒ f(x) is an odd function so

equation

equation

equation

⇒ g(-x) = g(x) ⇒ g(x) is an even function so

equation

Now given integral becomes

equation

equation

equation

 Divide  numerator and denominator by Cos2x we get

equation

equation

equation

Putting  tan x = t   ⇒   sec2x dx = dt  we get

equation

Integrating this and putting the limit we get

equation

Question 26
equation
Ans: 1
Solution Hint
equation 
equation 

equation

Question 27

equation
Ans:  equation
Question 28
equation 
equation

Solution Hint

Putting  sin x - cos x = t    (sin x + cos x)dx = dt

Squaring sin x - cos x = t on both side we get:  sin2x = 1- t2

Making all substitution we get

equation 

equation

equation 
Question 29

equation
Answer

equation 

Question 30

equation 
Solution : 

equation 

equation 

equation 
Question 31
Find  equation
Answer: Putting  1 + 2x = t2. then integrating w r t x we get 

equation  

THANKS FOR YOUR VISIT
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