NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.11
Page No 347:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_407b64a3.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_7f54b990.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_319681a6.gif)
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_54ab9afb.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_m67aa2b1f.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_664829e1.gif)
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_m423eccee.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_35986860.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_5ec97145.gif)
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_4a4c489.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_32989753.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_21644107.gif)
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_m5de665ca.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_5dbe20ae.gif)
It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_773d8ee3.gif)
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_6ad4c7e1.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_m9a4fcf0.gif)
It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_m20ebff4.gif)
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8330/NCERT_Solution_Math_Chapter_7_final_html_m3484e968.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8330/NCERT_Solution_Math_Chapter_7_final_html_478c6707.gif)
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8331/NCERT_Solution_Math_Chapter_7_final_html_6c69f563.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8331/NCERT_Solution_Math_Chapter_7_final_html_m23f2161a.gif)
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8332/NCERT_Solution_Math_Chapter_7_final_html_d58f318.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8332/NCERT_Solution_Math_Chapter_7_final_html_m26b91062.gif)
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_m236c3cc6.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_3eb02be4.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_32268a59.gif)
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_66e84c6b.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_mca9ceae.gif)
As sin2 (−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2x is an even function.
It is known that if f(x) is an even function, then ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_7237529.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_7237529.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_m1f03cb4.gif)
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m4f8e3f37.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m5ab190b8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m102ed4fd.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_3be073f7.gif)
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_1000a7cf.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_37730826.gif)
As sin7 (−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2x is an odd function.
It is known that, if f(x) is an odd function, then ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_m6052cec3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_m6052cec3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_m27c8fca6.gif)
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m53f198ae.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m2ee739fb.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m6cbd1067.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m18b59200.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m1b0fe3a3.gif)
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_mb6d95c9.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_1787c899.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_m26704fd1.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_42f38cbe.gif)
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2c1b6167.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_m17c379d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2d540b86.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_6ba7be51.gif)
sin (π − x) = sin x
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_71e40f52.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_5a0010e1.gif)
Adding (4) and (5), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2580c48b.gif)
Let 2x = t ⇒ 2dx = dt
When x = 0, t = 0 and when
x=π2, t=π∴
I=12∫0πlog sin tdt-π2log 2
⇒I=I2-π2log 2 [from 3]
⇒I2=-π2log 2
⇒I=-πlog 2
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_2ebf4e1d.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m17123bbd.gif)
It is known that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m4b151446.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m4b151446.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m67347065.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_7f16d75.gif)
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m61e0a3ca.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m7281926a.gif)
It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m41aba784.gif)
Question 19:
Show that
if f and g are defined as
and ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m2efbf4b0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_26f59125.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_4c24da22.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m2efbf4b0.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_3bb51308.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m34610f85.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m572b38bb.gif)
Question 20:
The value of
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_2ffa09b4.gif)
A. 0
B. 2
C. π
D. 1
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_1470d63d.gif)
It is known that if f(x) is an even function, then
and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_7237529.gif)
if f(x) is an odd function, then ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_m6052cec3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_m6052cec3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_m125c2688.gif)
Hence, the correct answer is C.
Question 21:
The value of
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_14873b9a.gif)
A. 2
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m2111158f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m2111158f.gif)
C. 0
D. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_395cc2cc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_395cc2cc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_30dd75fb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m28885fd1.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m42a37c68.gif)
Hence, the correct answer is C.