250+ TOP MCQs on Properties of Definite Integrals | Class 12 Maths

Mathematics Multiple Choice Questions & Answers (MCQs) on “Properties of Definite Integrals”.

1. What is the difference property of definite integrals?
a) (int_a^b)[-f(x)-g(x)dx
b) (int_a^b)[f(-x)+g(x)dx
c) (int_a^b)[f(x)-g(x)dx
d) (int_a^b)[f(x)+g(x)dx
Answer: c
Clarification: The sum difference property of definite integrals is (int_a^b)[f(x)-g(x)dx
(int_a^b)[f(x)-g(x)dx = (int_a^b)f(x)dx-(int_a^b)g(x)dx

2. The sum property of definite integrals is (int_a^b)[f(x)+g(x)dx?
a) False
b) True
Answer: b
Clarification: The sum property of definite integrals is (int_a^b)[f(x)+g(x)dx
(int_a^b)[f(x)+g(x)dx = (int_a^b)f(x)dx+(int_a^b)g(x)dx
Hence, it is true.

3. What is the constant multiple property of definite integrals?
a) (int_a^b)k⋅f(x)dy
b) (int_a^b)[f(-x)+g(x)dx
c) (int_a^b)k⋅f(x)dx
d) (int_a^b)[f(x)+g(x)dx
Answer: c
Clarification: The constant multiple property of definite integrals is (int_a^b)k⋅f(x)dx
(int_a^b)k⋅f(x)dx = k (int_a^b)f(x)dx

4. What is the reverse integral property of definite integrals?
a) –(int_a^b)f(x)dx=-(int_b^a)g(x)dx
b) –(int_a^b)f(x)dx=-(int_b^a)g(x)dx
c) (int_a^b)f(x)dx=(int_b^a)g(x)dx
d) (int_a^b)f(x)dx=-(int_b^a)f(x)dx
Answer: d
Clarification: In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is (int_a^b)f(x)dx=-(int_b^a)f(x)dx.

5. Identify the zero-length interval property.
a) (int_a^b)f(x)dx = -1
b) (int_a^b)f(x)dx = 1
c) (int_a^b)f(x)dx = 0
d) (int_a^b)f(x)dx = 0.1
Answer: c
Clarification: The zero-length interval property is one of the properties used in definite integrals and they are always positive. The zero-length interval property is (int_a^b)f(x)dx = 0.

6. What is adding intervals property?
a) (int_a^c)f(x)dx+(int_b^c)f(x)dx = (int_a^c)f(x) dx
b) (int_a^b)f(x)dx+(int_b^a)f(x)dx = (int_a^c)f(x) dx
c) (int_a^b)f(x)dx+(int_b^c)f(x)dx = (int_a^c)f(x) dx
d) (int_a^b)f(x)dx-(int_b^c)f(x)dx = (int_a^c)f(x) dx
Answer: c
Clarification: The adding intervals property of definite integrals is (int_a^b)f(x)dx+(int_b^c)f(x)dx.
(int_a^b)f(x)dx+(int_b^c)f(x)dx = (int_a^c)f(x) dx

7. What is the name of the property of (int_a^b)f(x)dx+(int_b^c)f(x)dx = (int_a^c)f(x) dx?
a) Zero interval property
b) Adding intervals property
c) Adding integral property
d) Adding integrand property
Answer: b
Clarification: (int_a^b)f(x)dx+(int_b^c)(x)dx = (int_a^c)f(x) dx is a property of definite integrals. (int_a^b)f(x)dx+(int_b^c)f(x)dx = (int_a^c)f(x) dx is called as adding intervals property used to combine a lower limit and upper limit of two different integrals.

8. What is the name of the property (int_a^b)f(x)dx=-(int_b^a)f(x)dx?
a) Reverse integral property
b) Adding intervals property
c) Zero interval property
d) Adding integrand property
Answer: a
Clarification: In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is (int_a^b)f(x)dx=-(int_b^a)f(x)dx.

9. What is the name of the property (int_a^b)f(x)dx = 0?
a) Reverse integral property
b) Adding intervals property
c) Zero-length interval property
d) Adding integrand property
Answer: b
Clarification: The zero-length interval property is one of the properties used in definite integrals and they are always positive. The zero-length interval property is (int_a^b)f(x)dx = 0.

10. What property this does this equation come under (int^1_{-1})sin⁡x dx=-(int_1^{-1})sin⁡x dx?
a) Reverse integral property
b) Adding intervals property
c) Zero-length interval property
d) Adding integrand property
Answer: a
Clarification: (int^1_{-1})sin⁡x dx=-(int_1^{-1})sin⁡x dx comes under the reverse integral property.
In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is (int_a^b)f(x)dx=-(int_b^a)f(x)dx.

11. Evaluate (int_2^3)3f(x)-g(x)dx, if (int_2^3)f(x) = 4 and (int_2^3)g(x)dx = 4.
a) 38
b) 12
c) 8
d) 7
Answer: c
Clarification: (int_2^3)3f(x)-g(x)dx = 3 (int_2^3)f(x) – (int_2^3)g(x)dx
= 3(4) – 4
= 8

12. Compute (int_3^2)f(x) dx if (int_2^3)f(x) = 4.
a) – 4
b) 84
c) 2
d) – 8
Answer: c
Clarification: (int_3^2)f(x)dx = – (int_2^3)f(x)dx
= – 4

13. Compute (int_8^2)2f(x)dx if (int_2^8)f(x) = – 3.
a) – 4
b) 84
c) 2
d) – 8
Answer: c
Clarification: (int_8^2)2f(x)dx = -2 (int_2^8)f(x)dx
= – 2(-3)
= 6

14. Compute (int_2^6)7e^x dx.
a) 30.82
b) 7(e⁶ – e²)
c) 11.23
d) 81(e⁶ – e³)
Answer: b
Clarification: (int_2^6)7e^x dx = 7(e^x)⁶₂ dx
= 7(e⁶ – e²)

15. Evaluate (int_3^7)2f(x)-g(x)dx, if (int_3^7)f(x) = 4 and (int_3^7)g(x)dx = 2.
a) 38
b) 12
c) 6
d) 7
Answer: c
Clarification: (int_3^7)2f(x)-g(x)dx = 2 (int_3^7)f(x) – (int_3^7)g(x)dx
= 2(4) – 2
= 6