NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.10
Page No 340:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_6d4f200d.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_26c4315c.gif)
When x = 0, t = 1 and when x = 1, t = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_mb530b8a.gif)
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_m4c0d7ecb.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_m4f40448a.gif)
Also, let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_132cbd71.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_132cbd71.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_e432d87.gif)
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m37dfca2b.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m5f5ba4a2.gif)
Also, let x = tanθ ⇒ dx = sec2θ dθ
When x = 0, θ = 0 and when x = 1, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_2fde353d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_2fde353d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_7416ad9b.gif)
Takingθas first function and sec2θ as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m1696ceaa.gif)
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m6b4ab1ea.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m4fa51e4.gif)
Let x + 2 = t2 ⇒ dx = 2tdt
When x = 0,
and when x = 2, t = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m7400b295.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_671afa1.gif)
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_m21d4da58.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_m21d4da58.gif)
Let cos x = t ⇒ −sinx dx = dt
When x = 0, t = 1 and when![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_ec9674e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_ec9674e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_7c4db91c.gif)
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m47f8aec.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m7f64c05b.gif)
Let
⇒ dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m4cfccca9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_6b4b32a9.gif)
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_m6af4a309.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_37652a3b.gif)
Let x + 1 = t ⇒ dx = dt
When x = −1, t = 0 and when x = 1, t = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_5fb9abe8.gif)
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_6979ef1b.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_6979ef1b.gif)
Let 2x = t ⇒ 2dx = dt
When x = 1, t = 2 and when x = 2, t = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_71eddcf4.gif)
Question 9:
The value of the integral
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_586429f9.gif)
A. 6
B. 0
C. 3
D. 4
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_590694d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_61485027.gif)
Let cotθ = t ⇒ −cosec2θ dθ= dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_1d3dd9f2.gif)
Hence, the correct answer is A.
Question 10:
If ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_m551acf9b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_m551acf9b.gif)
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_4bad8c8f.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_m7c5c373f.gif)
Hence, the correct answer is B.