Class 12 Mathematics Chapter 3 – Integration MCQs | FBISE | Grade 12 Math - MCQs

Practice these 50 exam-oriented MCQs from Chapter 3 – Integration for Class 12 Mathematics (FBISE). Topics include antiderivatives, indefinite and definite integrals, properties of definite integrals, Fundamental Theorem of Calculus, areas, volumes, trapezium rule, and applications of integration.

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1. The general antiderivative of $f(x)$ is a function $F(x)$ such that

A) $F'(x)=f(x)$ B) $F(x)=f'(x)$ C) $F'(x)=1/f(x)$ D) $F(x)=0$

2. If $F'(x)=f(x)$ , then $F(x)$ is called

A) the derivative of $f(x)$ B) an antiderivative of $f(x)$ C) the limit of $f(x)$ D) the inverse of $f(x)$

3. The indefinite integral $\int f(x)\,dx$ represents

A) only one numerical valueB) the derivative of $f(x)$ C) the family of antiderivatives of $f(x)$ D) the domain of $f(x)$

4. The constant $C$ in an indefinite integral is called the

A) constant of integrationB) integrandC) limit of integrationD) variable of integration

5. For $n e-1$ , $\int x^n\,dx$ equals

A) $nx^{n-1}+C$ B) $x^n/n+C$ C) $x^{n-1}/(n-1)+C$ D) $\frac{x^{n+1}}{n+1}+C$

6. $\int x^4\,dx$ is

A) $4x^3+C$ B) $\frac{x^5}{5}+C$ C) $x^4/4+C$ D) $5x^4+C$

7. $\int x^{-2}\,dx$ is

A) $-\frac1x+C$ B) $1/x+C$ C) $\ln|x|+C$ D) $-2x^{-3}+C$

8. $\int x^{-3}\,dx$ is

A) $-3x^{-4}+C$ B) $1/(2x^2)+C$ C) $-\frac{1}{2x^2}+C$ D) $-1/x^3+C$

9. $\int\sqrt{x}\,dx$ is

A) $2\sqrt{x}+C$ B) $\frac23x^{3/2}+C$ C) $x^{3/2}+C$ D) $x^{1/2}/2+C$

10. $\int k\,dx$ , where $k$ is constant, equals

A) $kx+C$ B) $k+C$ C) $x/k+C$ D) $kx$

11. $\int(3x^2-4x+5)\,dx$ is

A) $6x-4+C$ B) $3x^3-4x^2+5x+C$ C) $x^3-2x^2+5x+C$ D) $x^3-4x^2+5+C$

12. The constant multiple property is

A) $\int kf(x)dx=k\int f(x)dx$ B) $\int kf(x)dx=\int f(x)dx/k$ C) $\int kf(x)dx=kf(x)$ D) $\int kf(x)dx=k+\int f(x)dx$

13. The sum property gives

A) $\int(f+g)dx=\int fdx-\int gdx$ B) $\int(f+g)dx=\int fdx+\int gdx$ C) $\int(f+g)dx=f+g$ D) $\int(f+g)dx=\int fdx\int gdx$

14. $\int 5x^3\,dx$ is

A) $15x^2+C$ B) $20x^4+C$ C) $5x^4+C$ D) $\frac54x^4+C$

15. Which expression is an antiderivative of $2x$ ?

A) $x^2+C$ B) $2x^2+C$ C) $x+C$ D) $2+C$

16. A definite integral has

A) only a constant of integrationB) no integrandC) upper and lower limitsD) no variable of integration

17. In $\int_a^b f(x)\,dx$ , $a$ and $b$ are called

A) coefficientsB) limits of integrationC) antiderivativesD) constants of integration

18. In $\int_a^b f(x)\,dx$ , $f(x)$ is the

A) integrandB) upper limitC) constantD) antiderivative

19. The value of $\int_a^a f(x)\,dx$ is

A) $f(a)$ B) $1$ C) $2f(a)$ D) $0$

20. Reversing the limits gives

A) $\int_a^b fdx=\int_b^a fdx$ B) $\int_a^b fdx=-\int_b^a fdx$ C) both are always zeroD) their product

21. $\int_0^1 x\,dx$ equals

A) $1$ B) $1/2$ C) $2$ D) $0$

22. $\int_1^3 2x\,dx$ equals

A) $8$ B) $6$ C) $4$ D) $10$

23. $\int_0^2(x+1)\,dx$ equals

A) $3$ B) $2$ C) $4$ D) $5$

24. $\int_1^2x^2\,dx$ equals

A) $7/3$ B) $3$ C) $8/3$ D) $1$

25. $\int_0^\pi\sin x\,dx$ equals

A) $0$ B) $1$ C) $2$ D) $\pi$

26. The Fundamental Theorem of Calculus connects

A) matrices and determinantsB) differentiation and integrationC) probability and statisticsD) complex numbers and vectors

27. If $F'(x)=f(x)$ , then $\int_a^b f(x)dx$ equals

A) $F(a)+F(b)$ B) $F(a)F(b)$ C) $F(b)-F(a)$ D) $F(a)-F(b)$

28. A definite integral can represent

A) net areaB) only perimeterC) only slopeD) only volume

29. If a graph lies below the x-axis throughout an interval, its definite integral there is

A) positiveB) zeroC) negativeD) undefined

30. When $f(x)\ge0$ on $[a,b]$ , the area under $y=f(x)$ is

A) $f(b)-f(a)$ B) $\int_a^b f(x)dx$ C) $\int_b^a f(x)dx$ D) $f(a)f(b)$

31. The area under $y=x$ from $0$ to $3$ is

A) $3$ B) $6$ C) $9$ D) $9/2$

32. The area under $y=x^2$ from $0$ to $2$ is

A) $4$ B) $8/3$ C) $2$ D) $16/3$

33. The area between $y=x$ and $y=x^2$ from $0$ to $1$ is

A) $1/6$ B) $1/3$ C) $1/2$ D) $2/3$

34. For the area between two curves $y=f(x)$ and $y=g(x)$ , with $f(x)\ge g(x)$ , use

A) $\int_a^b[f(x)+g(x)]dx$ B) $\int_a^b[f(x)-g(x)]dx$ C) $\int_a^b g(x)dx$ D) $f(b)-g(a)$

35. The volume generated by rotating $y=f(x)$ about the x-axis is commonly found by

A) $2\pi\int f(x)dx$ B) $\int f(x)dx$ C) $\pi\int[f(x)]^2dx$ D) $\pi\int f(x)dx$

36. Rotating $y=x$ , $0\le x\le1$ , about the x-axis gives volume

A) $\pi/3$ B) $\pi$ C) $2\pi/3$ D) $1/3$

37. The trapezium rule is used to

A) find an exact derivativeB) estimate a definite integralC) solve every differential equationD) find a complex conjugate

38. For one trapezium of width $h$ and parallel sides $y_0,y_1$ , the area is

A) $h(y_0-y_1)/2$ B) $2h(y_0+y_1)$ C) $(y_0+y_1)/h$ D) $h(y_0+y_1)/2$

39. Using one trapezium, $\int_0^2x\,dx$ is exactly

A) $1$ B) $2$ C) $3$ D) $4$

40. The definite integral of a rate gives

A) accumulated changeB) only a derivativeC) only a limitD) only a domain

41. If velocity is $v(t)$ , then $\int_a^b v(t)dt$ gives

A) accelerationB) timeC) displacementD) jerk

42. If marginal cost is $C'(x)$ , integrating it over an interval gives

A) the domain of costB) change in total costC) the inverse of costD) the second derivative

43. If a growth rate is integrated over time, the result gives

A) accumulated changeB) instantaneous rateC) a derivativeD) a domain

44. If $f(x)$ changes sign on $[a,b]$ , $\int_a^b f(x)dx$ represents

A) always total geometric areaB) only positive areaC) signed net areaD) only negative area

45. To obtain total geometric area when a curve crosses the x-axis, one should

A) ignore the negative partB) use absolute values or split the intervalC) reverse both limitsD) differentiate the function

46. Which statement about an indefinite integral is correct?

A) It includes an arbitrary constant $C$ .B) It always has numerical limits.C) It is always one number.D) It cannot contain powers of $x$ .

47. Which statement about a definite integral is correct?

A) It always contains $+C$ .B) It has specified limits and evaluates to a numerical value.C) It has no integrand.D) It cannot represent area.

48. For an antiderivative $F(x)+C$ , changing $C$ does not change a definite integral because

A) the limits disappearB) the integrand becomes zeroC) the constants cancel in $F(b)-F(a)$ D) the variable disappears

49. If $\int f(x)\,dx=F(x)+C$ , then differentiating both sides gives

A) $F(x)+C$ B) $F'(x)$ C) $f(x)$ D) $C$

50. Which condition is necessary when using the power rule for $\int x^n dx$ ?

A) $n=0$ onlyB) $n e-1$ C) $n>1$ D) $n<0$ only

Explore More Class 12 Mathematics MCQs

Chapter 1 – Functions and Graphs Chapter 2 – Limit, Continuity and Derivative Chapter 3 – Integration Chapter 4 – Differential Equations Chapter 5 – Kinematics of Motion in a Straight Line Chapter 6 – Analytical Geometry Chapter 7 – Conic Section Chapter 8 – Inverse Trigonometric Functions and Their Graphs Chapter 9 – Solution of Trigonometric Equations Chapter 10 – Numerical Methods

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Syllabus & SLO Alignment

This set follows the FBISE HSSC-II Mathematics Assessment Framework for Curriculum 2022-23 and covers the Integration SLOs beginning with M-12-A-56, including general antiderivatives, properties and evaluation of integrals, definite integrals, the Fundamental Theorem of Calculus, area, volume, numerical integration and applications.

Reference: FBISE Final Assessment Framework & Model Question Paper, Mathematics HSSC-II, Curriculum 2022-23.

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Syllabus & SLO Alignment

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