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.