NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Integrals
NCERT Solutions for Class 12 Maths Chapter 7 contains all the questions with detailed solutions. Students are advised to solve these questions for better understanding of the concepts in chapter 7.
Krati Saraswat31 Jan, 2024
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NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise (Integrals)
NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Integrals is prepared by the academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of chapter-7. Given below is step by step solutions of all questions given in the NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise.NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Integrals Overview
NCERT Solutions for Class 12 Maths Chapter 7 covers these important topics. Students are encouraged to review each topic thoroughly in order to fully understand the concepts taught in the chapter and make optimal use of the provided solutions. These solutions are the outcome of the dedicated effort that the Physics Wallah teachers have been doing to aid students in understanding the ideas covered in this chapter. After going over and rehearsing these responses, the goal is for students to easily score outstanding exam results.NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise
Solve The Following Questions NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise: Integrate the function in Exercises 1 to 11. Question 1.
Solution :
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Question
2.
Solution :
.png)
Question
3.
.png)
Solution :
.png)
Taking
θ
as first function and sec
2
θ
as second function and integrating by parts, we obtain
.png)
Question
4.
Solution :
.png)
Question
5.
Solution :
.png)
Question
6.
Solution :
.png)
.png)
Question
7.
.png)
Solution :
.png)
Question
8.
.png)
Solution :
.png)
Question
9.
.png)
(A) 6
(B) 0
(C) 3
(D) 4
Solution :.png)
Let cot
θ
=
t
⇒ −cosec2
θ
d
θ
=
dt
.png)
Question
10.
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x
Solution : Let I =
Question
11.
Solution :
Integrate the function in Exercises 12 to 22.
Question
12.
Solution :
Question
13.
Solution :
Question
14.
Solution :
Question
15.
Solution :
Let I =
It can be seen that (
x
+ 2) ≤ 0 on [−5, −2] and (
x
+ 2) ≥ 0 on [−2, 5].
Question
16.
Solution :
Let I =
It can be seen that (
x
− 5) ≤ 0 on [2, 5] and (
x
− 5) ≥ 0 on [5, 8].
Question
17.
Solution :
Question
18.
Solution :
Question
19.
Solution :
Question
20.
Solution :
Question
21.
Solution :
Let I =
As sin
2
(−
x
) = (sin (−
x
))
2
= (−sin
x
)
2
= sin
2
x
, therefore, sin
2
x
is an even function.
Question
22.
Solution :
Evaluate the integrals in Exercises 23 and 24.
Question
23.
Solution :
Let I =
As sin
7
(−
x
) = (sin (−
x
))
7
= (−sin
x
)
7
= −sin
7
x
, therefore, sin
2
x
is an odd function.
Question
24.
Solution :
Evaluate the definite integrals in Exercise 25 to 33.
Question
25.
Solution :
Question
26.
Solution :
Adding (4) and (5), we obtain
Question
27.
Solution :
Question
28.
Solution :
Let I =
It can be seen that, (
x
− 1) ≤ 0 when 0 ≤
x
≤ 1 and (
x
− 1) ≥ 0 when 1 ≤
x
≤ 4
Question
29.Show that
if
f
and
g
are defined as f (x) = f(a - x) and g(x) + g(a - x) = 4
Solution :
Question
30.
A. 0
B. 2
C. π
D. 1
Solution :
= π
Question
31.
A.
2
B.
3/4
C.
0
D.
-2
Solution :
Question
32.
Solution :
From equation (1), we obtain
Question
33.
Solution :
Prove the following (Exercise 34 to 40).
Question
34.
[Hint: Put x = a/t]
Solution :
Question
35.
Solution :
Let I =
Question
36.
Solution :
Question
37.
Solution :
Question
38.
Solution :
Question
39.
Solution :
Question
40. Evaluate
as a limit of sum.
Solution :
Given:
It is known that,
Question
41. Choose the correct answer:
is equal to:
Solution :
Therefore, option (A) is correct.
Question
42. Choose the correct answer:
is equal to:
(A)
(B) log |sin x + cos x | + C
(C) log |sin x - cos x | + C
(D)
Solution :
Therefore, option (B) is correct.
Question
43.
Choose the correct answers If f (a + b – x) = f (x), then
Solution :
Therefore, option (D) is correct.
Question
44. The value of
is:
(A) 1
(B) 0
(C) -1
(D) π/4
Solution :
Therefore, option (B) is correct.
Solve The Following Questions.
Integrate the function in Exercises 1 to 11. Question 1.
Solution :
Question
2.
Solution :
Question
3.
Solution :
Taking
θ
as first function and sec
2
θ
as second function and integrating by parts, we obtain
Question
4.
Solution :
Question
5.
Solution :
Question
6.
Solution :
Question
7.
Solution :
Question
8.
Solution :
Question
9.
(A) 6
(B) 0
(C) 3
(D) 4
Solution :
Let cot
θ
=
t
⇒ −cosec2
θ
d
θ
=
dt
Question
10.
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x
Solution : Let I =
Question
11.
Solution :
Integrate the function in Exercises 12 to 22.
Question
12.
Solution :
Question
13.
Solution :
Question
14.
Solution :
Question
15.
Solution :
Let I =
It can be seen that (
x
+ 2) ≤ 0 on [−5, −2] and (
x
+ 2) ≥ 0 on [−2, 5].
Question
16.
Solution :
Let I =
It can be seen that (
x
− 5) ≤ 0 on [2, 5] and (
x
− 5) ≥ 0 on [5, 8].
Question
17.
Solution :
Question
18.
Solution :
Question
19.
Solution :
Question
20.
Solution :
Question
21.
Solution :
Let I =
As sin
2
(−
x
) = (sin (−
x
))
2
= (−sin
x
)
2
= sin
2
x
, therefore, sin
2
x
is an even function.
Question
22.
Solution :
Evaluate the integrals in Exercises 23 and 24.
Question
23.
Solution :
Let I =
As sin
7
(−
x
) = (sin (−
x
))
7
= (−sin
x
)
7
= −sin
7
x
, therefore, sin
2
x
is an odd function.
Question
24.
Solution :
Evaluate the definite integrals in Exercise 25 to 33.
Question
25.
Solution :
Question
26.
Solution :
Adding (4) and (5), we obtain
Question
27.
Solution :
Question
28.
Solution :
Let I =
It can be seen that, (
x
− 1) ≤ 0 when 0 ≤
x
≤ 1 and (
x
− 1) ≥ 0 when 1 ≤
x
≤ 4
Question
29.Show that
if
f
and
g
are defined as f (x) = f(a - x) and g(x) + g(a - x) = 4
Solution :
Question
30.
A. 0
B. 2
C. π
D. 1
Solution :
= π
Question
31.
A.
2
B.
3/4
C.
0
D.
-2
Solution :
Question
32.
Solution :
From equation (1), we obtain
Question
33.
Solution :
Prove the following (Exercise 34 to 40).
Question
34.
[Hint: Put x = a/t]
Solution :
Question
35.
Solution :
Let I =
Question
36.
Solution :
Question
37.
Solution :
Question
38.
Solution :
Question
39.
Solution :
Question
40. Evaluate
as a limit of sum.
Solution :
Given:
It is known that,
Question
41. Choose the correct answer:
is equal to:
Solution :
Therefore, option (A) is correct.
Question
42. Choose the correct answer:
is equal to:
(A)
(B) log |sin x + cos x | + C
(C) log |sin x - cos x | + C
(D)
Solution :
Therefore, option (B) is correct.
Question
43.
Choose the correct answers If f (a + b – x) = f (x), then
Solution :
Therefore, option (D) is correct.
Question
44. The value of
is:
(A) 1
(B) 0
(C) -1
(D) π/4
Solution :
Therefore, option (B) is correct.
NCERT Solutions for Class 12 Maths Chapter 7 FAQs
What is Chapter 7 of Class 12 Maths about?
Chapter 7 of Class 12 Maths is titled "Integration." It covers the concept of definite and indefinite integrals, properties of integrals, and various techniques of integration.
What is the Miscellaneous Exercise in Chapter 7 about?
The Miscellaneous Exercise in Chapter 7 includes additional problems related to integrals, providing students with more practice and a diverse range of questions.
Why is it important to study Integrals in Class 12 Maths?
Integrals are fundamental in calculus and are used to find the area under curves, solve problems related to accumulation of quantities, and many other applications in various fields.
How can NCERT Solutions for Chapter 7 Miscellaneous Exercise help students?
NCERT Solutions provide step-by-step explanations for each problem in the Miscellaneous Exercise. They help students understand the concepts, methods, and techniques used in solving a variety of integral problems.
Where can I find NCERT Solutions for Chapter 7 Miscellaneous Exercise?
NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise can be found in the official NCERT textbooks. Solutions may be available in various study materials, online educational platforms, or dedicated math solution books.
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