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On March 11, 2024, 11:35 AM

Exercise 7.4 - Chapter 7 Integrals class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7570/NCERT_Solution_Math_Chapter_7_final_html_m6d4a512a.gif

Answer:

Let x3 = t

∴ 3x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7570/NCERT_Solution_Math_Chapter_7_final_html_6c319a08.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7572/NCERT_Solution_Math_Chapter_7_final_html_m6eadee56.gif

Answer:

Let 2x = t

∴ 2dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7572/NCERT_Solution_Math_Chapter_7_final_html_2bb9ddb6.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7574/NCERT_Solution_Math_Chapter_7_final_html_m4f26274f.gif

Answer:

Let 2 − t

⇒ −dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7574/NCERT_Solution_Math_Chapter_7_final_html_m311edd7e.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7576/NCERT_Solution_Math_Chapter_7_final_html_12ee8f5b.gif

Answer:

Let 5x = t

∴ 5dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7576/NCERT_Solution_Math_Chapter_7_final_html_4152a2c9.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7577/NCERT_Solution_Math_Chapter_7_final_html_1313bb5.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7577/NCERT_Solution_Math_Chapter_7_final_html_521d393d.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7579/NCERT_Solution_Math_Chapter_7_final_html_686a690e.gif

Answer:

Let x3 = t

∴ 3x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7579/NCERT_Solution_Math_Chapter_7_final_html_687cb876.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m693dd56.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m6e073184.gif

From (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m5bd0fa9b.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7583/NCERT_Solution_Math_Chapter_7_final_html_1f129e0a.gif

Answer:

Let x3 = t

⇒ 3x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7583/NCERT_Solution_Math_Chapter_7_final_html_m163da311.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7584/NCERT_Solution_Math_Chapter_7_final_html_m46808fa3.gif

Answer:

Let tan x = t

∴ sec2x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7584/NCERT_Solution_Math_Chapter_7_final_html_654e25c6.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7586/NCERT_Solution_Math_Chapter_7_final_html_m94a3620.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7586/NCERT_Solution_Math_Chapter_7_final_html_m518e3476.gif

Question 11:

19×2+6x+5

Answer:

∫19×2+6x+5dx=∫13x+12+22dx

 

Let (3x+1)=t

3 dx=dt

 

⇒∫13x+12+22dx=13∫1t2+22dt

 

=13×2tan-1t2+C

 

=16tan-13x+12+C

 

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7589/NCERT_Solution_Math_Chapter_7_final_html_168db442.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7589/NCERT_Solution_Math_Chapter_7_final_html_10a5934e.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7591/NCERT_Solution_Math_Chapter_7_final_html_62650cce.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7591/NCERT_Solution_Math_Chapter_7_final_html_m7b6ec913.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7593/NCERT_Solution_Math_Chapter_7_final_html_62f95dc1.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7593/NCERT_Solution_Math_Chapter_7_final_html_m60221a62.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7596/NCERT_Solution_Math_Chapter_7_final_html_m67d6b929.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7596/NCERT_Solution_Math_Chapter_7_final_html_m1d0caab4.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_664b5d67.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_m1b1241f3.gif

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

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_2a4a7359.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_39c73fad.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_b662a70.gif

Equating the coefficients of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_m668374ed.gif

From (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_1ddb53d5.gif

From equation (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_7731a74.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m59186f15.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m72fdaa74.gif

Equating the coefficient of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m1c9032eb.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m2bfaa834.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_1ab063c8.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m192518e9.gif

Substituting equations (2) and (3) in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m4ebced4d.gif

Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m9c52b48.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_6a0bbc55.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m5ae3a8ee.gif

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

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m597fb40f.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m55743fdf.gif

Substituting equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m151359b3.gif

Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_3a973c5a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m1e4b7ad4.gif

Equating the coefficients of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m68eba682.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_17bfe748.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m70c40c54.gif

Question 21:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_4ba941c2.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m5aaab5ed.gif

Let x2 + 2x +3 = t

⇒ (2x + 2) dx =dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_1869b74b.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m53d4ecad.gif https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m67495a4.gif

Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_m36d7c819.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_7848b5f8.gif

Equating the coefficients of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_4b12c12d.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_m713d5001.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_68d718b3.gif

Substituting (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_6aab90c.gif

Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_17e7b92.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_445b6746.gif

Equating the coefficients of x and constant term, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m58eeddc7.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m224dadf8.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m479038b2.gif

Question 24:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7629/NCERT_Solution_Math_Chapter_7_final_html_492a94c1.gif equals

A. x tan−1 (x + 1) + C

B. tan− 1 (x + 1) + C

C. (x + 1) tan−1 x + C

D. tan−1 x + C

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7629/NCERT_Solution_Math_Chapter_7_final_html_m59b21429.gif

Hence, the correct Answer is B.

Question 25:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m55ee7506.gif equals

A. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_3c4d7613.gif

B. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m2edc7a68.gif

C. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_7a2f9c65.gif

D. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m7ff80582.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m563bf5a5.gif

Hence, the correct Answer is B.

Also Read : Exercise-7.5-Chapter-7-Integrals-class-12-ncert-solutions-Maths

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