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NCERT Solutions for Class 12th Mathematics-II

 

Page 4 of 12

Chapter 7. Integrals

Exercise 7.4

 

 

 

Exercise 7.4

Integrate the following functions in Exercises 1 to 23.

Ques.1.  

Ans. Let I = 

 ……….(i)

Putting 

  

 

  From eq. (i),

I = 

Ques.2.  

Ans. 

Ques.3.  

Ans. 

Ques.4.  

Ans. 

Ques.5.  

Ans. Let I = 

 ……….(i)

Putting 

  

 

  From eq. (i),

I = 

Ques.6.  

Ans. Let I = 

 ……….(i)

Putting 

 

 

From eq. (i),

I = 

 = 

Ques.7.  

Ans. Let I = 

Let I1 = 

Putting 

 

 

 I1 =  =  =  = 

 I = 

 I =  where 

Ques.8.  

Ans. Let I = 

……….(i)

Putting 

  

 

  From eq. (i),

I = 

Ques.9.  

Ans. Let I = ……….(i)

Putting 

 

 

  From eq. (i),

I = 

Ques.10.  

Ans. 

Ques.11.  

Ans. 

[For making completing the squares]

Ques.12.  

Ans. 

Ques.13. 

Ans. 

=  

=  

Ques.14.  

Ans. 

=  

=  

Ques.15.  

Ans. 

Ques.16.  

Ans. Let I =  ……….(i)

Putting 

 

 

 From eq. (i),

I = 

Ques.17.  

Ans. Let I = 

 …….(i)

Let I1 = 

Putting 

 

I1 = 

Putting this value in eq. (i),

Ques.18.  

Ans. Let I =  ……….(i)

Let Linear = A  (Quadratic) + B

 ……(ii)

 

Comparing coefficients of  6A = 5

 A = 

Comparing constants,

2A + B = 

On solving, we get A = , B = 

Putting the values of A and B in eq. (ii),

Putting this value of  in eq. (i),

I = 

I = 

 I =  ……….(iii)

Now I1 = 

Putting 

 

 

I1 =  =  =  ……….(iv)

Again I2 =

 ……….(v)

Putting values of I1 and I2 in eq. (iii),

I = 

Ques.19. 

Ans. Let I = 

 ……….(i)

Let Linear = A  (Quadratic) + B

  ……(ii)

 

Comparing coefficients of  2A = 6  A = 3

Comparing constants, –9A + B = 7

On solving, we get A = 3, B = 34

Putting the values of A and B in eq. (ii),

Putting this value of  in eq. (i),

I = 

I = 

 I = ……….(iii)

Now I1 = 

Putting 

 

 

I1 = 

 = 

 ……….(iv)

Again I2 =

 ……….(v)

Putting values of I1 and I2 in eq. (iii),

I = 

Ques.20.  

Ans. Let I =  ……….(i)

Let Linear = A  (Quadratic) + B

 ……(ii)

 

Comparing coefficients of  –2A = 1  A = 

Comparing constants, 4A + B = 2

On solving, we get A = B = 4

Putting the values of A and B in eq. (ii),

Putting this value of  in eq. (i),

I = 

I = 

 I =  ……….(iii)

Now I1 = 

Putting 

 

 

 I1 =  = 

 = 

…….(iv)

Again I2 =

……….(v)

Putting values of I1 and I2 in eq. (iii),

I = 

Ques.21.  

Ans. Let I = 

Let Linear = A  (Quadratic) + B

 ……(ii)

 

Comparing coefficients of  2A = 1  A = 

Comparing constants, 2A + B = 2

On solving, we get A = B = 1

Putting the values of A and B in eq. (ii),

Putting this value of  in eq. (i),

I = 

I = 

 I =  ……….(iii)

Now I1 = 

Putting 

 

 

I1 =  = 

 = 

 ……….(iv)

Again I2 =

 ……….(v)

Putting values of I1 and I2 in eq. (iii),

I = 

Ques.22.  

Ans. Let I =  ……….(i)

Let Linear = A  (Quadratic) + B

 ……(ii)

 

Comparing coefficients of  2A = 1

 A = 

Comparing constants, –2A + B = 3

On solving, we get A = B = 4

Putting the values of A and B in eq. (ii),  

Putting this value of  in eq. (i),

I = 

I = 

 I =  ……….(iii)

Now I1 = 

Putting 

 

 

I1 =  = 

 ……….(iv)

Again I2 =

……….(v)

Putting values of I1 and I2 in eq. (iii),

I = 

Ques.23 

Ans. Let I = ……….(i)

Let Linear = A  (Quadratic) + B

 

  ……(ii)

 

Comparing coefficients of  2A = 5  A = 

Comparing constants, 4A + B = 2

On solving, we get A = B = 

Putting the values of A and B in eq. (ii),

Putting this value of  in eq. (i),

I = 

I = 

 I = ……….(iii)

Now I1 = 

Putting 

 

 

I1 = 

 = 

……….(iv)

Again I2 =

 ……….(v)

Putting values of I1 and I2 in eq. (iii),

I = 

Choose the correct answer in Exercise 24 and 25.

Ques.24.  equals

(A)  

(B)  

(C)  

(D) 

Ans. 

hance, option (B) is correct.

Ques.25.  equals

(A) 

(B) 

(C)  

(D) 

Ans. Let I = 

 

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