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# NCERT Solutions class 12 Maths Exercise 7.4

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## NCERT Solutions class 12 Maths Integrals

Integrate the following functions in Exercises 1 to 9.

1.

Ans. Let I =

=  ……….(i)

Putting

From eq. (i),

I =

=

=

2.

Ans.

=

=

=

3.

Ans.

=

=

=

4.

Ans.

=

=

=

5.

Ans. Let I =

=  ……….(i)

Putting

From eq. (i),

I =

=

=

=

=

6.

Ans. Let I =

=

=  ……….(i)

Putting

From eq. (i),

I =

=

=

=

7.

Ans. Let I =

=

=

=

Let I1 =

Putting

I1 =  =  =  =

I =

=

I =  where

8.

Ans. Let I =

= ……….(i)

Putting

From eq. (i),

I =

=

=

=

9.

Ans. Let I = ……….(i)

Putting

From eq. (i),

I =

=

=

=

Integrate the following functions in Exercises 10 to 18.

10.

Ans.

=

=

=

=

11.

Ans.

=

=

=

[For making completing the squares]

=

=

=

=

=

=

12.

Ans.

=

=

=

=

=

=

=

13.

Ans.

=

=

=

=

14.

Ans.

=

=

=

=

=

15.

Ans.

=

=

=

=

=

=

=

=

=

16.

Ans. Let I =  ……….(i)

Putting

From eq. (i),

I =

=

=

=

=

17.

Ans. Let I =

=

=

=  …….(i)

Let I1 =

=

Putting

I1 =

=

=

=

=

Putting this value in eq. (i),

### 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 =

### Integrate the following functions in Exercises 19 to 23.

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 =

### 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 =

### 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 =

### 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 =

### 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.

24.  equals

(A)

(B)

(C)

(D)

Ans.

=

=

=

Therefore, option (B) is correct.

### 25.  equals

(A)

(B)

(C)

(D)

Ans. Let I =

=

=

=

=

=

=

=

=

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Exercise 7.4

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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