NCERT Solutions class 12 Maths Exercise 7.2 (Ex 7.2) Chapter 7 Integrals


NCERT Solutions for Class 12 Maths Exercise 7.2 hapter 7 Integrals – FREE PDF Download

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NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.2) Exercise 7.2



Integrate the functions in Exercise 1 to 8.

1. 

Ans. Putting  

 

 


2. 

 

Ans. Putting  

 

 

 


3. 

 

Ans. Putting  

 

 

 


4. 

 

Ans. Putting  

 

 

 

 = 


5. 

 

Ans.  

12sin2(ax+b)dx12∫sin⁡2(ax+b)dx

12[cos(2ax+2b)2a]+c12[−cos⁡(2ax+2b)2a]+c                     [  becauseSin(ax+b)dx=1aCos(ax+b)∫Sin(ax+b)dx=−1aCos(ax+b)]


6. 

 

Ans.  

Using  (ax+b)ndx=(ax+b)n+1a(n+1)+c∫(ax+b)ndx=(ax+b)n+1a(n+1)+c  We have


7. 

 

Ans.  

Using  (ax+b)ndx=(ax+b)n+1a(n+1)+c∫(ax+b)ndx=(ax+b)n+1a(n+1)+c

 


8. 

 

Ans. Let I =  

 ……….(i)

Putting 

 

From eq. (i),

I = 


Integrate the functions in Exercise 9 to 17.

 

9. 

Ans. Let I =  

 …..(i)

Putting 

 

 

 From eq. (i), I = 


10. 

 

Ans. Let I = ……….(i) 

Putting 

 

 

 

From eq. (i),

I = 


11. 

 

Ans.  

          Using  (ax+b)ndx=(ax+b)n+1a(n+1)+c∫(ax+b)ndx=(ax+b)n+1a(n+1)+c

2x+4−−−−√(x+434)+c2x+4(x+43−4)+c

2x+4−−−−√(x+4123)+c2x+4(x+4−123)+c


12. 

 

Ans. Let I =  

 ……….(i)

Putting 

 

 

From eq. (i), I = 


13. 

 

Ans. Let I =  

……….(i)

Putting 

 

From eq. (i), I = 


14. 

 

Ans. Let I =  

……….(i)

Putting 

 

 

From eq. (i), I =  = 


15. 

 

Ans. Let I =  

……….(i)

Putting 

 

 

From eq. (i), I =  = 


16. 

 

Ans.  

         Usingeax+bdx=eax+ba+c∫eax+bdx=eax+ba+c


17. 

 

Ans. Let I =  

……….(i

Putting 

 

 

From eq. (i), I =  = 

using   eax+bdx=eax+ba+c∫eax+bdx=eax+ba+c

We have   =12(et1)+c=12(e−t−1)+c


Integrate the functions in Exercise 18 to 26.

 

18. 

Ans. Let I =  ……….(i) 

Putting 

 

From eq. (i), I = 


19. 

 

Ans. Let I =  

 [Multiplying each term by ]

Putting 

 

 

From eq. (i), I =  = 


20. 

 

Ans. Let I =  

……….(i)

Putting 

 

 

 

From eq. (i), I = 


21. 

 

Ans.  

Using  sec2(ax+b)dx=tan(ax+b)a+c∫sec2(ax+b)dx=tan⁡(ax+b)a+c

=tan(2x3)2x+c=tan⁡(2x−3)2−x+c


22. 

 

Ans.  

Using   sec2(ax+b)dx=tan(ax+b)a+c∫sec2(ax+b)dx=tan⁡(ax+b)a+c


23. 

 

Ans. Let I = ……….(i) 

Putting 

 

 

From eq. (i), I = 


24. 

 

Ans. Let I =  

……….(i)

Putting 

 

 

From eq. (i), I =  = 


25. 

 

Ans. Let I =  

……….(i)

Putting 

 

From eq. (i),  I =  = 


26. 

 

Ans. Let I = ……….(i) 

Putting 

 

 

 

From eq. (i),  I = 

=


Integrate the functions in Exercise 27 to 37.

 

27. 

Ans. Let I =  

 ……….(i)

Putting 

 

 

From eq. (i),  I = 


28. 

 

Ans. Let I = ……….(i) 

Putting 

 

 

From eq. (i),  I = 


29. 

 

Ans. Let I =  ……….(i) 

Putting 

 

 

 

From eq. (i),  I = 


30. 

 

Ans. Let I =  

 ……….(i)

Putting 

 

 

From eq. (i),  I = 


31. 

 

Ans. Let I =  

……….(i)

Putting 

 

 

From eq. (i),  I = 


32. 

 

Ans. Let I =  

Adding and subtracting  in the numerator,

 = 

where  ……….(i)

Putting 

 

 

I1 =  = 

Putting this value in eq. (i), we get required integral,


33. 

 

Ans. Let I =  

Adding and subtracting  in the numerator,

12cosxsinxcosxsinx+sinx+cosxcosxsinxdx12∫cos⁡x−sin⁡xcos⁡x−sin⁡x+sin⁡x+cos⁡xcos⁡x−sin⁡xdx

12(1+sinx+cosxcosxsinx)dx12∫(1+sin⁡x+cos⁡xcos⁡x−sin⁡x)dx


34. 

 

Ans. Let I =  

…..(i)

Putting 

 

From eq. (i), I = 


35. 

 

Ans. Let I = ……….(i) 

Putting 

 

 

From eq. (i), I = 


36. 

 

Ans. Let I =  ……….(i) 

Putting 

 

 

 

From eq. (i), I = 


37. 

 

Ans. Let I =  

 ……….(i)

Putting 

 

 

From eq. (i), I = 


Choose the correct answer in Exercise 38 and 39.

 

38.  equals

(A) 

(B) 

(C) 

(D) log(10x+x10)+clog(10x+x10)+c

Ans. Let I =  ……….(i) 

Putting 

 

From eq. (i), I = 

Therefore, option (D) is correct.


39.  equals

 

(A) 

(B) 

(C) 

(D) 

Ans.  

Therefore, option (B) is correct.