NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.3
Page No 307:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7510/NCERT_Solution_Math_Chapter_7_final_html_158e4d81.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7510/NCERT_Solution_Math_Chapter_7_final_html_m25ad4f6f.gif)
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_m79d80b67.gif)
Answer:
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_e048a77.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_e048a77.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_4e6849bf.gif)
Question 3:
cos 2x cos 4x cos 6x
Answer:
It is known that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7523/NCERT_Solution_Math_Chapter_7_final_html_mf795f95.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7523/NCERT_Solution_Math_Chapter_7_final_html_mf795f95.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7523/NCERT_Solution_Math_Chapter_7_final_html_m453c4158.gif)
Question 4:
sin3 (2x + 1)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_m634ea88b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_m634ea88b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_2257b9f2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_m5795a85f.gif)
Question 5:
sin3 x cos3 x
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7527/NCERT_Solution_Math_Chapter_7_final_html_m73e75233.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7527/NCERT_Solution_Math_Chapter_7_final_html_69753c1b.gif)
Question 6:
sin x sin 2x sin 3x
Answer:
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7529/NCERT_Solution_Math_Chapter_7_final_html_4735bd7c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7529/NCERT_Solution_Math_Chapter_7_final_html_4735bd7c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7529/NCERT_Solution_Math_Chapter_7_final_html_4b01ff05.gif)
Question 7:
sin 4x sin 8x
Answer:
It is known that,
sin A . sin B = 12cosA-B-cosA+B∴∫sin4x sin8x dx=∫12cos4x-8x-cos4x+8xdx=12∫cos-4x-cos12xdx=12∫cos4x-cos12xdx=12sin4x4-sin12x12+C
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7533/NCERT_Solution_Math_Chapter_7_final_html_4d6e72c3.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Kayako_FKY%2019079408.gif)
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7535/NCERT_Solution_Math_Chapter_7_final_html_7705fbfd.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7535/NCERT_Solution_Math_Chapter_7_final_html_m316e357f.gif)
Question 10:
sin4 x
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7536/NCERT_Solution_Math_Chapter_7_final_html_m48648a1c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7536/NCERT_Solution_Math_Chapter_7_final_html_m31f5f804.gif)
Question 11:
cos4 2x
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7538/NCERT_Solution_Math_Chapter_7_final_html_m2f3cf2d6.gif)
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7540/NCERT_Solution_Math_Chapter_7_final_html_639207d4.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7540/NCERT_Solution_Math_Chapter_7_final_html_1b7154f9.gif)
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7543/NCERT_Solution_Math_Chapter_7_final_html_m3bf74025.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7543/NCERT_Solution_Math_Chapter_7_final_html_m3df62037.gif)
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7544/NCERT_Solution_Math_Chapter_7_final_html_m2f3df7c7.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7544/NCERT_Solution_Math_Chapter_7_final_html_m3ed05535.gif)
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7547/NCERT_Solution_Math_Chapter_7_final_html_61e38253.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7547/NCERT_Solution_Math_Chapter_7_final_html_m3ab061dc.gif)
Question 16:
tan4x
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7549/NCERT_Solution_Math_Chapter_7_final_html_m781709fe.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7549/NCERT_Solution_Math_Chapter_7_final_html_m15daa35b.gif)
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7552/NCERT_Solution_Math_Chapter_7_final_html_390003e3.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7552/NCERT_Solution_Math_Chapter_7_final_html_m4da9ac1c.gif)
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7554/NCERT_Solution_Math_Chapter_7_final_html_m4c746474.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7554/NCERT_Solution_Math_Chapter_7_final_html_2529779e.gif)
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7557/NCERT_Solution_Math_Chapter_7_final_html_m473f7a6d.gif)
Answer:
1sinxcos3x=sin2x+cos2xsinxcos3x=sinxcos3x+1sinxcosx
⇒1sinxcos3x=tanxsec2x+1cos2xsinxcosxcos2x=tanxsec2x+sec2xtanx
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/zafar1111.png)
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7559/NCERT_Solution_Math_Chapter_7_final_html_3ac37af4.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7559/NCERT_Solution_Math_Chapter_7_final_html_m57da292d.gif)
Question 21:
sin−1 (cos x)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_5ca185ec.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_34c59b7.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_67f31dd1.gif)
Substituting in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_62bbad50.gif)
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7564/NCERT_Solution_Math_Chapter_7_final_html_m299efddf.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7564/NCERT_Solution_Math_Chapter_7_final_html_m6d95357.gif)
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7566/NCERT_Solution_Math_Chapter_7_final_html_m356fefc4.gif)
A. tan x + cot x + C
B. tan x + cosec x + C
C. − tan x + cot x + C
D. tan x + sec x + C
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7566/NCERT_Solution_Math_Chapter_7_final_html_m5794483e.gif)
Hence, the correct answer is A.
Question 24:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m28f9787.gif)
A. − cot (exx) + C
B. tan (xex) + C
C. tan (ex) + C
D. cot (ex) + C
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m46251d8.gif)
Let exx = t
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m410bc16d.gif)
Hence, the correct answer is B.