NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.2
Page No 304:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_2daa2dc2.gif)
Answer:
Let
= t
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_44a7d5a8.gif)
∴2x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_285bfdcb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_4f3f798c.gif)
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_4fda87ef.gif)
Answer:
Let log |x| = t
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_52ef2a81.gif)
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m61551e94.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m4370fbad.gif)
Let 1 + log x = t
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m2ebe9270.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_1cd43b92.gif)
Question 4:
sin x ⋅ sin (cos x)
Answer:
sin x ⋅ sin (cos x)
Let cos x = t
∴ −sin x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7408/NCERT_Solution_Math_Chapter_7_final_html_69fb365b.gif)
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_42180c1b.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_1400fe22.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_m6b8fb0e5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_m6b8fb0e5.gif)
∴ 2adx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_396f0375.gif)
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m6a9188e.gif)
Answer:
Let ax + b = t
⇒ adx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m5727662a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m10c63328.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m31dc4753.gif)
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_6cb9b43.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_m5d0e67fa.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_m5d0e67fa.gif)
∴ dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_4c24a780.gif)
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7416/NCERT_Solution_Math_Chapter_7_final_html_4e827515.gif)
Answer:
Let 1 + 2x2 = t
∴ 4xdx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7416/NCERT_Solution_Math_Chapter_7_final_html_m34de4fbb.gif)
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m4b12630c.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m7c0ef6b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m7c0ef6b.gif)
∴ (2x + 1)dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_76f6bdb0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m75db78a0.gif)
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_520c9935.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_4563a36f.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_m3aa49064.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_m3aa49064.gif)
∴![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_mf4b4123.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_mf4b4123.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_2d5d2240.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_1a4ba108.gif)
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7422/NCERT_Solution_Math_Chapter_7_final_html_44d0f94f.gif)
Answer:
Let I=∫xx+4 dxput x+4=t⇒dx=dt
Now, I=∫t-4tdt=∫t-4t-1/2dt=23t3/2-42t1/2+C=23.t.t1/2-8t1/2+C=23x+4x+4-8x+4+C
=23xx+4+83x+4-8x+4+C=23xx+4-163x+4+C
=23x+4x-8+C
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_m132ed99c.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_6e628098.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_6e628098.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_27cd530.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_27cd530.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_m452c22f6.gif)
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_3d95f002.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_26539378.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_26539378.gif)
∴ 9x2 dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_m649abaa8.gif)
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_78cb1997.gif)
Answer:
Let log x = t
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_3e19e712.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_a99ad2.gif)
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m42ca294.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m7516177.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m7516177.gif)
∴ −8x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m15aecbfc.gif)
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_2939b26a.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_5bcc50a3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_5bcc50a3.gif)
∴ 2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_m513791a7.gif)
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_6ddfda2d.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_m2b929949.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_m2b929949.gif)
∴ 2xdx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_m650f6bc9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_7c9ba91.gif)
Page No 305:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m481a6bf9.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m62a1bac4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m62a1bac4.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m26ae7d6c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m26ae7d6c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_277fdce0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m666c0c5.gif)
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_4bdb887f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_4bdb887f.gif)
Dividing numerator and denominator by ex, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_m6cb48a41.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_c22a63c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_c22a63c.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_549b934c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_549b934c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_m449ce1c0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_79d625cd.gif)
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m8dd01a5.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m67028847.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m67028847.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m3d96998a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m3d96998a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m3314c039.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m5a357383.gif)
Question 21:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_60990151.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_60990151.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_50852ccc.gif)
Let 2x − 3 = t
∴ 2dx = dt
⇒∫tan22x-3dx = ∫sec22x-3 – 1dx=∫sec2t- 1dt2= 12∫sec2t dt – ∫1dt= 12tant – t + C= 12tan2x-3 – 2x-3 + C
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7458/NCERT_Solution_Math_Chapter_7_final_html_30c46dae.gif)
Answer:
Let 7 − 4x = t
∴ −4dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7458/NCERT_Solution_Math_Chapter_7_final_html_5dbac121.gif)
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_59d50245.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_m127d1393.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_m127d1393.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_45979e45.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_45979e45.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_d934bc5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_m615ee39c.gif)
Question 24:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_m1e861f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_m1b300ce0.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_1d11b643.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_1d11b643.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_6bb65edd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_6bb65edd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_29ad5f75.gif)
Question 25:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_m67c449bc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_6fe644c1.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_m589b30d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_m589b30d9.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_ad32e6e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_ad32e6e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_56990d0f.gif)
Question 26:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_m3f7ab82.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_m7e7680c5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_m7e7680c5.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_mf4b4123.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_mf4b4123.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_1195f5d9.gif)
Question 27:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_396850ae.gif)
Answer:
Let sin 2x = t
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_ad1a990.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_ad1a990.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_103e3be3.gif)
Question 28:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_m24b03964.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_589aee2d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_589aee2d.gif)
∴ cos x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_4631f4db.gif)
Question 29:
cot x log sin x
Answer:
Let log sin x = t
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7485/NCERT_Solution_Math_Chapter_7_final_html_m3e62cda.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7485/NCERT_Solution_Math_Chapter_7_final_html_8ef846c.gif)
Question 30:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7487/NCERT_Solution_Math_Chapter_7_final_html_5910cc29.gif)
Answer:
Let 1 + cos x = t
∴ −sin x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7487/NCERT_Solution_Math_Chapter_7_final_html_537be00f.gif)
Question 31:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7489/NCERT_Solution_Math_Chapter_7_final_html_594e5e18.gif)
Answer:
Let 1 + cos x = t
∴ −sin x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7489/NCERT_Solution_Math_Chapter_7_final_html_m24732181.gif)
Question 32:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_m70797b4a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_42b7f736.gif)
Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_e9f2ce0.gif)
Question 33:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_70acf21f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_m1fecaf67.gif)
Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_1c1841d8.gif)
Question 34:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7494/NCERT_Solution_Math_Chapter_7_final_html_m3cd2a5e1.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7494/NCERT_Solution_Math_Chapter_7_final_html_m441fc4b5.gif)
Question 35:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_6ef9e05f.gif)
Answer:
Let 1 + log x = t
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_m79272764.gif)
Question 36:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_663e2a5d.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_m7e821b61.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_5eda337c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_5eda337c.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_12ec54d0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_12ec54d0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_m2f765048.gif)
Question 37:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_3d29e0a5.gif)
Answer:
Let x4 = t
∴ 4x3 dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m4c5a1b01.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m3d91c446.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m3d91c446.gif)
∴![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_4d47be1c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_4d47be1c.gif)
From (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_626f7efb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m776b9de8.gif)
Question 38:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_3cbaf76f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_21be7c9a.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_4e59dfbc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_4e59dfbc.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_m65c6fb2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_m65c6fb2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_m3b428ede.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_3b394c63.gif)
Hence, the correct answer is D.
Question 39:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m52ea234.gif)
A. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m57a70745.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m57a70745.gif)
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_eb8e735.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_eb8e735.gif)
C. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m2169ce25.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m2169ce25.gif)
D. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m296cab0a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m296cab0a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m3109d37e.gif)
Hence, the correct answer is B.