NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.4
Page No 315:
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 − x = 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)
Page No 316:
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)
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)
A. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_3c4d7613.gif)
![](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)
![](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)
![](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)
![](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.