12 maths ncertsolutions chapter 7 4
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(Chapter – 7) (Integrals)
(Class – XII)
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1
Exercise 7.4
Integrate the functions in Exercises 1 to 23.
Question 1:
Answer 1:
Let x3 = t
3x2 dx = dt
Question 2:
Answer 2:
Let 2x = t
2dx = dt
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Question 3:
Answer 3:
Let 2 − x = t
⇒ −dx = dt
Question 4:
Answer 4:
Let 5x = t
5dx = dt
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Question 5:
Answer 5:
Question 6:
Answer 6:
Let x3 = t
3x2 dx = dt
![Page 4: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/4.jpg)
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Question 7:
Answer 7:
From (1), we obtain
Question 8:
Answer 8:
Let x3 = t
⇒ 3x2 dx = dt
![Page 5: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/5.jpg)
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Question 9:
Answer 9:
Let tan x = t
∴ sec2x dx = dt
Question 10:
Answer 10:
![Page 6: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/6.jpg)
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Question 11:
Answer 11:
Question 12:
Answer 12:
![Page 7: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/7.jpg)
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Question 13:
Answer 13:
![Page 8: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/8.jpg)
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Question 14:
Answer 14:
![Page 9: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/9.jpg)
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Question 15:
Answer 15:
![Page 10: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/10.jpg)
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Question 16:
Answer 16:
Equating the coefficients of x and constant term on both sides, we obtain
4A = 4 ⇒ A = 1
A + B = 1 ⇒ B = 0
Let 2x2 + x − 3 = t
∴ (4x + 1) dx = dt
Question 17:
Answer 17:
Equating the coefficients of x and constant term on both sides, we obtain
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From (1), we obtain
From equation (2), we obtain
Question 18:
Answer 18:
Equating the coefficient of x and constant term on both sides, we obtain
![Page 12: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/12.jpg)
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Substituting equations (2) and (3) in equation (1), we obtain
![Page 14: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/14.jpg)
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Question 19:
Answer 19:
Equating the coefficients of x and constant term, we obtain
2A = 6 ⇒ A = 3
−9A + B = 7 ⇒ B = 34
∴ 6x + 7 = 3 (2x − 9) + 34
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Substituting equations (2) and (3) in (1), we obtain
Question 20:
Answer 20:
Equating the coefficients of x and constant term on both sides, we obtain
![Page 16: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/16.jpg)
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Using equations (2) and (3) in (1), we obtain
![Page 17: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/17.jpg)
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Question 21:
Answer 21:
Let x2 + 2x +3 = t
⇒ (2x + 2) dx =dt
![Page 18: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/18.jpg)
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Using equations (2) and (3) in (1), we obtain
Question 22:
Answer 22:
Equating the coefficients of x and constant term on both sides, we obtain
![Page 19: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/19.jpg)
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Substituting (2) and (3) in (1), we obtain
Question 23:
Answer 23:
Equating the coefficients of x and constant term, we obtain
![Page 20: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/20.jpg)
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Using equations (2) and (3) in (1), we obtain
![Page 21: 12 Maths NcertSolutions Chapter 7 4](https://reader035.vdocuments.net/reader035/viewer/2022062316/577c83ee1a28abe054b6e029/html5/thumbnails/21.jpg)
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Question 24: equals
(A) x tan–1 (x + 1) + C (B) tan–1 (x + 1) + C
(C) (x + 1) tan–1x + C (D) tan–1x + C
Answer 24:
Hence, the correct Answer is B.
Question 25: equals
(A) (B)
(C) (D)
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Answer 25:
Hence, the correct Answer is B.