Class 12 Maths MCQ – Integrals

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1. The integral ∫x dx is:

a) (x^2)/2 + C
b) x^2 + C
c) x + C
d) 1 + C

Answer:

a) (x^2)/2 + C

Explanation:

The power of x increases by 1 and then we divide by the new power.

2. The integral of a constant 'c' is:

a) x + C
b) c + C
c) cx + C
d) 0

Answer:

c) cx + C

Explanation:

The integration of a constant is the constant times x plus the arbitrary constant C.

3. The integral ∫(x^2 + 3x + 2) dx is:

a) (x^3)/3 + (3x^2)/2 + 2x + C
b) (x^2)/2 + 3x + 2 + C
c) x^3 + 3x^2 + 2x + C
d) (x^3)/3 + x^2 + 2x + C

Answer:

a) (x^3)/3 + (3x^2)/2 + 2x + C

Explanation:

Each term is integrated separately using the power rule.

4. The integral ∫cos(x) dx is:

a) sin(x) + C
b) cos(x) + C
c) -sin(x) + C
d) -cos(x) + C

Answer:

a) sin(x) + C

Explanation:

The integral of cos(x) is sin(x).

5. Which of the following is an indefinite integral?

a) ∫x dx from 0 to 1
b) ∫(x^2 + 1) dx
c) ∫sin(x) dx from π/2 to π
d) ∫e^x dx from 1 to 2

Answer:

b) ∫(x^2 + 1) dx

Explanation:

Indefinite integrals do not have specified limits of integration.

6. The fundamental theorem of calculus connects:

a) Limit and continuity
b) Differentiation and algebra
c) Differentiation and integration
d) Algebra and geometry

Answer:

c) Differentiation and integration

Explanation:

The theorem connects the concepts of differentiation and integration.

7. The integral ∫sec^2(x) dx is:

a) tan(x) + C
b) cot(x) + C
c) sec(x) + C
d) csc(x) + C

Answer:

a) tan(x) + C

Explanation:

The integral of sec^2(x) is tan(x).

8. Which symbol represents the integral operation?

a) Δ
b) ∑
c) π
d) ∫

Answer:

d) ∫

Explanation:

The symbol ∫ is used to represent integration.

9. The antiderivative of a function f(x) is given by:

a) f'(x)
b) ∫f(x) dx
c) f''(x)
d) Δf(x)

Answer:

b) ∫f(x) dx

Explanation:

Antiderivative is another name for the indefinite integral.

10. ∫(d/dx of f(x)) dx is:

a) f(x) + C
b) f'(x) + C
c) f''(x) + C
d) 0

Answer:

a) f(x) + C

Explanation:

Integrating the derivative of a function gives back the original function.

11. The area under the curve y = f(x) between x = a and x = b is given by:

a) ∫f(x) dx from a to b
b) ∫f'(x) dx from a to b
c) ∫f''(x) dx from a to b
d) ∫f(x) dx from b to a

Answer:

a) ∫f(x) dx from a to b

Explanation:

The definite integral gives the area under the curve between two limits.

12. The integral ∫sin^2(x) dx is:

a) (x – sin(x)cos(x))/2 + C
b) (x + sin(x)cos(x))/2 + C
c) (x/2) – (sin(2x)/4) + C
d) (x/2) + (sin(2x)/4) + C

Answer:

c) (x/2) – (sin(2x)/4) + C

Explanation:

Use the identity sin^2(x) = (1 – cos(2x))/2 and then integrate.

13. If f'(x) = g(x), then ∫g(x) dx is:

a) f(x)
b) f(x) + C
c) g(x)
d) g(x) + C

Answer:

b) f(x) + C

Explanation:

The integral of g(x) will be an antiderivative of f(x).

14. The integral of a sum is the sum of the integrals. This property is called:

a) Linearity of integration
b) Integration by parts
c) Substitution rule
d) Chain rule

Answer:

a) Linearity of integration

Explanation:

This property is fundamental to the linearity of the integral operation.

15. The method used to integrate the product of two functions is:

a) Linearity of integration
b) Integration by parts
c) Substitution rule
d) Chain rule

Answer:

b) Integration by parts

Explanation:

Integration by parts is a technique used for the product of two functions.

16. The integral ∫e^x dx is:

a) e^x + C
b) ln(x) + C
c) e^(-x) + C
d) 1/e^x + C

Answer:

a) e^x + C

Explanation:

The integral of e^x is itself e^x.

17. The integral ∫ln(x) dx is:

a) xln(x) – x + C
b) xln(x) + C
c) x/ln(x) + C
d) ln^2(x)/2 + C

Answer:

a) xln(x) – x + C

Explanation:

The integration of ln(x) yields xln(x) minus x.

18. If ∫f(x) dx = F(x) + C, then F(x) is:

a) The definite integral of f
b) A family of functions
c) The antiderivative of f
d) The derivative of f

Answer:

c) The antiderivative of f

Explanation:

F(x) represents the antiderivative of the function f(x).

19. The integral ∫(1/x) dx, x > 0, is:

a) x + C
b) 1/x + C
c) ln(x) + C
d) e^x + C

Answer:

c) ln(x) + C

Explanation:

The integral of 1/x for positive x is the natural logarithm of x.

20. The area between two curves y = f(x) and y = g(x) from x = a to x = b is given by:

a) ∫(f(x) – g(x)) dx from a to b
b) ∫(g(x) – f(x)) dx from a to b
c) Either a) or b), depending on which function is on top
d) ∫f(x)g(x) dx from a to b

Answer:

c) Either a) or b), depending on which function is on top

Explanation:

The area is the difference of the two functions, but the order depends on which function lies above the other in the given interval.

Related MCQ (Multiple Choice Questions) :

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