NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.6
Page No 327:
Question 1:
x sin x
Answer:
Let I = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7675/NCERT_Solution_Math_Chapter_7_final_html_16f819ca.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7675/NCERT_Solution_Math_Chapter_7_final_html_16f819ca.gif)
Taking x as first function and sin x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7675/NCERT_Solution_Math_Chapter_7_final_html_m5b52456b.gif)
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m565b8df6.gif)
Answer:
Let I = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m242fdeeb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m242fdeeb.gif)
Taking x as first function and sin 3x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m214c7ae3.gif)
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_45da2d8.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_5ecce068.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_5ecce068.gif)
Taking x2 as first function and ex as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_m1406435d.gif)
Again integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_243d97bb.gif)
Question 4:
x logx
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7678/NCERT_Solution_Math_Chapter_7_final_html_m78a8c8a4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7678/NCERT_Solution_Math_Chapter_7_final_html_m78a8c8a4.gif)
Taking log x as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7678/NCERT_Solution_Math_Chapter_7_final_html_m5b95e34d.gif)
Question 5:
x log 2x
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7679/NCERT_Solution_Math_Chapter_7_final_html_3c936f78.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7679/NCERT_Solution_Math_Chapter_7_final_html_3c936f78.gif)
Taking log 2x as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7679/NCERT_Solution_Math_Chapter_7_final_html_m50a12fc3.gif)
Question 6:
x2 log x
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7680/NCERT_Solution_Math_Chapter_7_final_html_m50234465.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7680/NCERT_Solution_Math_Chapter_7_final_html_m50234465.gif)
Taking log x as first function and x2 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7680/NCERT_Solution_Math_Chapter_7_final_html_126181e4.gif)
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m70b6b8fb.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m7753b72d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m7753b72d.gif)
Taking
as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m17c7f48.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m3a459646.gif)
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_11d5bd80.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_m475935f3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_m475935f3.gif)
Taking
as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_m1a8b5181.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_m19e79690.gif)
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_5ad5a2f.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_7b3b2f71.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_7b3b2f71.gif)
Taking cos−1 x as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_1991c6d7.gif)
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_2aa59a77.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_37a9f912.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_37a9f912.gif)
Taking
as first function and 1 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_2aa59a77.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_783e653d.gif)
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m7beafb64.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_c428cc9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_c428cc9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m54c945f9.gif)
Taking
as first function and
as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m47731349.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m5f93d4cd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_56d7e816.gif)
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_4e99fa3a.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_5977bb13.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_5977bb13.gif)
Taking x as first function and sec2x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_m384594e7.gif)
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m1a8b5181.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_226da9cf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_226da9cf.gif)
Taking
as first function and 1 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m1a8b5181.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m4bf69e04.gif)
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_541373dc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_m783b9696.gif)
Taking
as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_m4b6918a2.gif)
I=log x 2∫xdx-∫ddxlog x 2∫xdxdx=x22log x 2-∫2log x .1x.x22dx=x22log x 2-∫xlog x dx
Again integrating by parts, we obtain
I = x22logx 2-log x ∫x dx-∫ddxlog x ∫x dxdx=x22logx 2-x22log x -∫1x.x22dx
=x22logx 2-x22log x +12∫x dx=x22logx 2-x22log x +x24+C
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m7e288453.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m7d7d9059.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m7d7d9059.gif)
Let I = I1 + I2 … (1)
Where,
and ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m3afbd7eb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_5adba482.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m3afbd7eb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_508ac40a.gif)
Taking log x as first function and x2 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_64161ae9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m3afbd7eb.gif)
Taking log x as first function and 1 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m33301466.gif)
Using equations (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_32c47506.gif)
Page No 328:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_m16dbffa1.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_m3fc69b14.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_m3fc69b14.gif)
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_8ec6dc1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_8ec6dc1.gif)
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_2d6339dd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_2d6339dd.gif)
∴ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_264157a1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_264157a1.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_ma0bf94e.gif)
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_41e8a549.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_f2746bb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_f2746bb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_356f9e57.gif)
Let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m311bf58a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_4bcecbea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m311bf58a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m692fc956.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m234a6149.gif)
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_2d60c21c.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_1b8fb60a.gif)
Let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_23e4f06.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_23c6706d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_23e4f06.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_m483f6657.gif)
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_425f49ab.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_509a61e0.gif)
Also, let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m123a8371.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m47d9edac.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m123a8371.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m7a0cc9fc.gif)
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m5d04b491.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m1ffb9b2d.gif)
Let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m4e016495.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_2cc6670b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m4e016495.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m3379bca.gif)
Question 21:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m198871a2.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m349755a1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m349755a1.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m72845c80.gif)
Again integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m624ee5cc.gif)
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_5c4b1fa6.gif)
Answer:
Let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m5b0e6d85.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_2effc955.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m5b0e6d85.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_2c176593.gif)
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_1772d35e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m2a375b8f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_1772d35e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m2a375b8f.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m2f12cf1c.gif)
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_36e73ee8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_m2073483f.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_56dd90a3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_56dd90a3.gif)
Also, let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_27cd530.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_770f1019.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_27cd530.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_m3e91749d.gif)
Hence, the correct answer is A.
Question 24:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_46f2088a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_42378067.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_46f2088a.gif)
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_594a5f7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_594a5f7.gif)
Also, let
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_m7b826a5e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_m1536fdf4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_m7b826a5e.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_31dddd75.gif)
Hence, the correct answer is B.