250+ TOP MCQs on Definite Integral | Class 12 Maths

Mathematics Multiple Choice Questions on “Definite Integral”.

1. In (int_a^b)f(y) dy, what is ‘a’ called as?
a) Integration
b) Upper limit
c) Lower limit
d) Limit of an integral
Answer: b
Clarification: In (int_a^b)f(y) dy ‘a’ is the called as lower limit and ‘b’ is called the upper limit of the integral. The fuction f in (int_a^b)f(y) dy is called the integrand. The letter ‘y’ is a dummy symbol and can be replaced by any other symbol.

2. The value of (int_0^{pi })sin ⁡y dy is 2.
a) True
b) False
Answer: a
Clarification: (int_0^{pi })sin⁡ y dy = [-cos y]^y₀
= – cos π – (- cos 0)
= -(-1)-(-1)
= 2

3. Compute ∫cos(x)-(frac {3}{x4})dx.
a) sin(x)+(frac {3}{4})x^-7+c
b) sec(x)+(frac {3}{4})x^-3+c
c) sin(x)+(frac {3}{4})x^-3
d) sin(x)+(frac {3}{4})x^-3+c
Answer: d
Clarification: ∫cos(x)-(frac {3}{x4})dx = ∫cos(x) dx−∫3(x^-4) dx
= sin(x)+(frac {3}{4})x^-3+c

4. What is the value of (int_2^3)cos⁡(x)-(frac {3}{x4})dx .
a) sin (3) – sin (2)
b) sin (3) – sin (9) – (frac {19}{288})
c) sin (8) – sin (2) – (frac {19}{288})
d) sin (3) – sin (2) – (frac {19}{288})
Answer: d
Clarification: (int_2^3)cos⁡(x)-(frac {3}{x4})dx = (int_2^3)sin(x) dx + (int_2^3 frac {3}{4})x^-3 dx
= (sin (3) + (frac {3}{4})3^-3) – (sin (2) + (frac {3}{4})2^-3)
= sin (3) – sin (2) – (frac {19}{288})

5. Evaluate (int_7^9)cos⁡(x)dx.
a) 8 (-sin 9 – sin 7)
b) 8 (sin 9 + sin 7)
c) 8 (sin 9 – sin 7)
d) 7 (sin 9 – sin 7)
Answer: c
Clarification: (int_7^9)8cos⁡(x)dx = 8 (int_7^9)cos⁡(x)dx
= 8 (cos x)⁹₇
= 8 (sin 9 – sin 7)

6. Compute (int_2^3 frac {cos⁡x-sin⁡x}{4})dx.
a) (frac {1}{4}) (sin 2 + cos 3 – sin 3 – cos 2)
b) (frac {1}{4}) (sin 3 – cos 3 – sin 2 – cos 2)
c) (frac {1}{4}) (sin 3 + cos 3 – sin 2 – cos 2)
d) (frac {1}{4}) (sin 3 + cos 3 + sin 2 – cos 2)
Answer: c
Clarification: (int_2^3 frac {cos⁡x-sin⁡x}{4})dx = (frac {1}{4}) [sin x – (- cos x)]³₂
= (frac {1}{4}) (sin x + cos x)³₂
= (frac {1}{4}) (sin 3 + cos 3) – (frac {1}{4}) (sin 2 + cos 2)
= (frac {1}{4}) (sin 3 + cos 3 – sin 2 – cos 2)

7. What is y in (int_a^b)f(y) dy called as?
a) Random variable
b) Dummy symbol
c) Integral
d) Integrand
Answer: b
Clarification: In (int_a^b)f(y) dy ‘a’ is the called as lower limit and ‘b’ is called the upper limit of the integral. The fuction ‘f’ in (int_a^b)f(y) dy is called the integrand. The letter ‘y’ is a dummy symbol and can be replaced by any other symbol.

8. The value of (int_1^2)1y⁵ dy is_____
a) 10.5
b) 56
c) 9
d) 23
Answer: a
Clarification: (int_1^2)1y⁵ dy = (y⁶/6)²₁
= (frac {64}{6} – frac{1}{6})
= 10.5

9. The value of (int_1^2)1y⁵/5dy is _____
a) 12
b) 2.1
c) 21
d) 11.1
Answer: b
Clarification: (int_1^2)1y⁵/5dy = (frac {1}{5})(y⁶/6)²₁
= (frac {1}{5}(frac {64}{6} – frac{1}{6}))
= 2.1

10. Evaluate (int_0^{pi })sin⁡x dx.
a) 2
b) 6
c) 17
d) 3
Answer: a
Clarification: (int_0^{pi })sin⁡x dx = [- cos x]^x₀
= – cos π – (-cos 0)
= -(-1)-(-1)
= 2

11. Evaluate (int_2^3)cosx dx.
a) 38.2
b) sin (9) – sin (4)
c) 89.21
d) sin (3) – sin (2)