NCERT Solutions class 12 Maths Exercise 7.5

NCERT Solutions class 12 Maths Exercise 7.5 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

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NCERT Solutions Class 12 Maths Integrals

 

Integrate the (rational) function in Exercises 1 to 6.

1. 

Ans.  …….(i)

 

 

Comparing coefficients of  on both sides A + B = 1 …..(ii)

Comparing constants 2A + B = 0 …..(iii)

Solving eq. (ii) and (iii), we get A =  and B = 2

Putting these values of A and B in eq. (i),

 

=−log|x+1|+2log|x+2|+c=−log|x+1|+2log|x+2|+c

= 

= 

---

2. 

Ans. 

= 

= 

= 

---

3. 

Ans. 

=  …….(i)

 

 

 

Comparing coefficients of : A + B + C = 0 …….(ii)

Comparing coefficients of : –5A – 4B – 3C= 3

 5A + 4B + 3C = –3 …….(iii)

Comparing constants: 6A + 3B + 2C = –1 …….(iv)

On solving eq. (i), (ii) and (iii), we get A = 1, B = –5, C = 4

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

= 

 

= 

 

= 

---

4. 

Ans. 

=  …….(i)

 

 

 

Comparing coefficients of : A + B + C = 0 …….(ii)

Comparing coefficients of : –5A – 4B – 3C= 1

 5A + 4B + 3C = –1 …….(iii)

Comparing constants: 6A + 3B + 2C = 0 …….(iv)

On solving eq. (i), (ii) and (iii), we get A =  B = –2, C = 

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

= 

 

= 

 

= 

---

5. 

Ans. 

= 

= 

= 

=  ….(i)

 

 

Comparing coefficients of  on both sides A + B = 2 …….(ii)

Comparing constants 2A + B = 0 …..(iii)

Solving eq. (ii) and (iii), we get A =  and B = 4

Putting these values of A and B in eq. (i),

 

=−2log|x+1|+4log|x+2|+c=−2log|x+1|+4log|x+2|+c

=4log|x+2|−2log|x+1|+c=4log|x+2|−2log|x+1|+c

---

 

6. 

Ans. 

= 

= 

=  (Dividing numerator by denominator)

 

=  …….(i)

Now 

=  …….(ii)

 

 

Comparing coefficients of  on both sides –2A + B =  …..(iii)

Comparing constants A = 1 …….(iv)

Solving eq. (ii) and (iii), we get A =  and B = 

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

 

= 

= 

Putting this value in eq. (i),

= 

---

Integrate the following function in Exercises 7 to 12.

7. 

Ans.  …….(i)

 

 

Comparing coefficients of  A + C = 0 …….(ii)

Comparing coefficients of , –A + B = 1 …….(iii)

Comparing constant terms, –B + C = 0 …….(iv)

Solving eq. (ii), (iii) and (iv), we get A =  B =  and C = 

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

 

= 

 

 

 

 

---

8. 

Ans. 

=  …….(i)

 

 

 

Comparing coefficients of : A + C = 0 …….(ii)

Comparing coefficients of : A + B – 2C= 1 …….(iii)

Comparing constants: –2A + 2B + C = 0 …….(iv)

On solving eq. (i), (ii) and (iii), we get

A =  B =  C = 

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

= 

 

= 

= 

= 

= 

= 

---

9. 

Ans. 

= 

= 

= 

= 

=  …….(i)

 

 

 

Comparing coefficients of : A + C = 0 …….(ii)

Comparing coefficients of : B – 2C= 3 …….(iii)

Comparing constants: –2A + B + C = 5 …….(iv)

On solving eq. (i), (ii) and (iii), we get A =  B = 4, C = 

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

= 

= 

= 

= 

= 

---

10. 

Ans. 

= 

=  …….(i)

 

 

 

Comparing coefficients of : 2A + 2B + C = 0 …….(ii)

Comparing coefficients of : 5A + B = 2 …….(iii)

Comparing constants: 3A – 3B – C = –3 …….(iv)

On solving eq. (i), (ii) and (iii), we get A =  B =  C = 

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

= 

  = 

= 

= 

=52log|x+1|−110log|x−1|−125log|2x+3|+c=52log|x+1|−110log|x−1|−125log|2x+3|+c

---

11. 

Ans. 

= 

=  …….(i)

 

 

 

Comparing coefficients of : A + +B + C = 0 …….(ii)

Comparing coefficients of : –B + 3C= 5 …….(iii)

Comparing constants: –4A – 2B + 2C = 0 …….(iv)

On solving eq. (i), (ii) and (iii), we get A =  B =  C = 

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

= 

= 

=53log|x+1|−52log|x+2|+56log|x−2|+c=53log|x+1|−52log|x+2|+56log|x−2|+c

---

12. 

Ans. 

=  …….(i)

[On dividing numerator by denominator]

Let 

= 

=  …….(ii)

 

 

Comparing coefficients of : A + B = 2 …….(iii)

Comparing constants: –A + B = 1 …….(iv)

On solving eq. (iii) and (iv), we get A =  B = 

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

= 

Putting this value in eq. (i),

= 

 

= 

= 

---

Integrate the following function in Exercises 13 to 17.

13. 

Ans. 

=  …….(i)

  = 

  = 

Comparing the coefficients of  A – B = 0 …….(ii)

Comparing the coefficients of  B – C = 0 …….(iii)

Comparing constants A + C = 2 …….(iv)

On solving eq. (ii), (iii) and (iv), we get A = 1, B = 1, C = 1

Putting these values of A, B and C in eq. (i),

 = 

=  = 

= 

= 

= 

---

14. 

Ans. Let I =  …….(i)

Putting 

 

 

 

Putting this value in eq. (i),

I = 

= 

= 

= 

= 

= 

= 

= 

= 

---

15. 

Ans. 

= 

Putting , 

=  …..(i)

 

 

Comparing the coefficients of  A + B = 0 ……(ii)

Comparing constants A – B = 1 …….(iii)

On solving the eq. (ii) and (iii), we get A =  B = 

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

 

= 

= 

---

16. 

Ans. Let I = 

Multiplying both numerator and denominator by ,

I = 

=  ……..(i)

Putting 

 

 

 From eq. (i),

I = 

= 

= 

= 

= 

= 

= 

= 

---

17. 

Ans. Let I =  …….(i)

Putting 

 

 

 From eq. (i), I = 

= 

= 

= 

= 

= 

= 

= 

= 

---

Integrate the following function in Exercises 18 to 21.

18. 

Ans.  …….(i)

Putting 

=  …….(ii)

Dividing numerator by denominator,

=  ….(iii)

Let  …….(iv)

 

 

Comparing coefficients of  A + B = –4 …….(v)

Comparing constants 4A + 3B = –10 …….(vi)

On solving eq. (v) and (vi), we get A = 2, B = –6

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

= 

= 

= 

= 

= 

---

19. 

Ans. Let I =  …….(i)

Putting   

 From eq. (i),

I = 

= 

= 

= 

= 

= 

= 

= 

= 

---

20. 

Ans. Let I = 

= 

= …(i)

Putting 

 

 

Putting this value in eq. (i),

I = 

= 

I = 

= 

= 

= 

= 

= 

---

21. 

Ans. Let I =  …….(i)

Putting 

 

 

 

 From eq. (i),

I = 

= 

= 

= 

= 

= 

= 

= 

= 

= 

---

Choose the correct answer in each of the Exercise 22 and 23.

22.  equals:

(A) 

(B) 

(C) 

(D) 

Ans. Let  …….(i)

 

 

Comparing coefficients of  A + B = 1 …….(ii)

Comparing constants –2A – B = 0 …….(iii)

On solving eq. (ii) and (iii), we get A = –1, B = 2

Putting these values of A and B in eq. (i),

 

= 

= 

= 

Therefore, option (B) is correct.

---

23.  equals:

(A) 

(B) 

(C) 

(D) 

Ans. Let I = 

= 

=  …(i)

Putting 

 

 

Putting this value in eq. (i),

I = 

= 

I = 

= 

= 

= 

= 

= 

Therefore, option (A) is correct.

NCERT Solutions class 12 Maths Exercise 7.5

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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