Integration

Maths, HSSC-Punjab Second Year. This chapter is split into 91 topics and has 2,521 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Definite integrals of exponential and logarithmic functions 30 questions
2. Rational expressions of e^x 29 questions
3. Volumes by cross-sections and standard solids 30 questions
4. Standard integral formulas (table) 26 questions
5. Logarithmic rule: derivative of the denominator in the numerator 30 questions
6. Substitution with logarithms and inverse trigonometric functions 25 questions
7. Partial fractions: repeated linear factors 30 questions
8. Expanding or simplifying before integrating 30 questions
9. Completing the square in the denominator 30 questions
10. Area between two curves (lines and parabolas) (set 2) 25 questions
11. Moment of inertia by integration 30 questions
12. Integration by parts: formula and choice of u and dv 22 questions
13. Initial value problems with trigonometric, exponential and rational rates 30 questions
14. Definite integrals by parts 30 questions
15. Substitution with exponential functions 30 questions
16. Integration by parts - x times exponential or trigonometric function 28 questions
17. Areas bounded by curves of the form x = f(y) 21 questions
18. Even and odd integrands over symmetric intervals 20 questions
19. Integrals of 1/sqrt(a^2 + x^2), 1/sqrt(a^2 - x^2) and 1/sqrt(x^2 - a^2) 30 questions
20. Volumes of solids of revolution about the x-axis 30 questions
21. Partial fractions: coefficients for distinct linear factors 30 questions
22. Integrating rational functions with irreducible quadratic factors 30 questions
23. Reciprocal and root of a linear expression 30 questions
24. Integrating rational functions with distinct linear factors 30 questions
25. Area between two curves (lines and parabolas) (set 1) 27 questions
26. Moment of inertia, average value and area under a concentration curve: ideas 24 questions
27. Indefinite integral notation and terminology 21 questions
28. Second-order rates: two conditions fix two constants 30 questions
29. Derivative of an integral with variable limits 30 questions
30. Definite integrals needing trigonometric identities 30 questions
31. Completing the square under a root sign 30 questions
32. Velocity, displacement and distance from acceleration 30 questions
33. Finding f(x) from f'(x) and one value 30 questions
34. Substitution with square roots of expressions 30 questions
35. Definite integral: definition and fundamental theorems 21 questions
36. Volumes of solids of revolution about the y-axis 30 questions
37. Integrals of the form e^x[f(x) + f'(x)] 20 questions
38. Trigonometric substitution: cases and special integrals 23 questions
39. Trigonometric substitution: results of the substitutions 28 questions
40. Antiderivatives and their families 24 questions
41. Substitution with powers of trigonometric functions 30 questions
42. Signed area versus total area 28 questions
43. Consumer and producer surplus from curved demand and supply 30 questions
44. Logarithmic rule with trigonometric and exponential functions 30 questions
45. Trigonometric identities before integrating 30 questions
46. Position, velocity and free fall by integration (set 2) 21 questions
47. Area under a curve above the x-axis 29 questions
48. Definite integrals of polynomials 29 questions
49. Sums of standard trigonometric integrals 30 questions
50. Rational functions needing division before partial fractions 28 questions
51. Surplus with linear demand and supply, and total surplus 30 questions
52. Consumer and producer surplus: definitions and formulas 22 questions
53. Average value of a function 30 questions
54. Exponential functions of a linear argument 30 questions
55. Surplus by integrating with respect to price 28 questions
56. Integrals of sqrt(a^2 - x^2), sqrt(x^2 + a^2) and sqrt(x^2 - a^2) 30 questions
57. Method of substitution: theorem and idea 21 questions
58. Integrals of e^(ax) sin bx and e^(ax) cos bx 30 questions
59. Position, velocity and free fall by integration (set 1) 21 questions
60. Partial fractions: cases and forms of decomposition 20 questions
61. Population and growth-rate problems 30 questions
62. General power rule with polynomial bases 30 questions
63. Checking an antiderivative by differentiation 30 questions
64. Piecewise and absolute-value integrands 30 questions
65. Integrals of 1/(a^2 - x^2), 1/(x^2 - a^2) and 1/(a^2 + x^2) 30 questions
66. Area between a parabola or cubic and the x-axis 25 questions
67. Integrating rational functions with repeated linear factors 30 questions
68. Using f(x) + f(-x) and periodic or symmetric conditions 29 questions
69. Symmetry property: f(x) / (f(x) + f(a + b - x)) 25 questions
70. Area under a curve: formulas and signed area 21 questions
71. Solids of revolution and volumes by cross-sections: formulas 20 questions
72. Properties of definite integrals: statements and uses 20 questions
73. Integrals over [0, 2a] and [0, pi] using f(2a - x) 27 questions
74. Integration by parts applied twice 29 questions
75. Integrating polynomials by the power rule 30 questions
76. Products of sines and cosines (sum formulas) 30 questions
77. Integration by parts - logarithmic and inverse trigonometric functions 21 questions
78. Integrals leading to inverse trigonometric functions 30 questions
79. Integrals of a^x and e^x combinations 29 questions
80. Definite integrals of trigonometric functions 30 questions
81. Basic rules of integration 24 questions
82. Using given values of definite integrals (additivity and linearity) 30 questions
83. Power rule with negative and fractional exponents 30 questions
84. Definite integrals of roots, reciprocals and powers 30 questions
85. Definite integrals by substitution (new limits) 30 questions
86. Drug concentration and area under the curve 30 questions
87. Powers of a linear expression (linear substitution) 30 questions
88. Definite integrals giving inverse trigonometric and logarithmic values 30 questions
89. Areas with modulus functions and ratios of areas 30 questions
90. Trigonometric functions of a linear argument 30 questions
91. Partial fractions: irreducible quadratic factors 30 questions

Sample questions: Integration

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Find the total area enclosed between the curve y = x³ − 2x² − 5x + 6 and the x-axis.

• A. 63/2
• B. 253/24
• C. 253/12(correct answer)
• D. 265/12

Answer: C.The x-intercepts are −2, 1, 3. Integrate |f| over each interval between consecutive intercepts and add: 253/12 square units; the signed integral over the whole span would be 125/12 because parts below the axis count negatively.

Q2. Find the total area enclosed between the curve y = x³ − 2x² − 3x and the x-axis.

• A. −45/4
• B. 71/3
• C. 71/6(correct answer)
• D. 65/6

Answer: C.The x-intercepts are −1, 0, 3. Integrate |f| over each interval between consecutive intercepts and add: 71/6 square units; the signed integral over the whole span would be −32/3 because parts below the axis count negatively.

Q3. Find the total area enclosed between the curve y = x³ − 5x² + 2x + 8 and the x-axis.

• A. 253/12(correct answer)
• B. 63/2
• C. 253/6
• D. 265/12

Answer: A.The x-intercepts are −1, 2, 4. Integrate |f| over each interval between consecutive intercepts and add: 253/12 square units; the signed integral over the whole span would be 125/12 because parts below the axis count negatively.

Q4. Find the total area enclosed between the curve y = x³ − 3x² + 2x and the x-axis.

• A. 0
• B. 1/4
• C. 1/2(correct answer)
• D. 1

Answer: C.The x-intercepts are 0, 1, 2. Integrate |f| over each interval between consecutive intercepts and add: 1/2 square units; the signed integral over the whole span would be 0 because parts below the axis count negatively.

Q5. Find the total area enclosed between the curve y = x³ − 8x² + 15x and the x-axis.

• A. 241/12
• B. 265/12
• C. 253/12(correct answer)
• D. 253/24

Answer: C.The x-intercepts are 0, 3, 5. Integrate |f| over each interval between consecutive intercepts and add: 253/12 square units; the signed integral over the whole span would be 125/12 because parts below the axis count negatively.

Q6. Find the total area enclosed between the curve y = x³ + 2x² − 3x and the x-axis.

• A. 71/6(correct answer)
• B. 32/3
• C. −7/12
• D. 45/2

Answer: A.The x-intercepts are −3, 0, 1. Integrate |f| over each interval between consecutive intercepts and add: 71/6 square units; the signed integral over the whole span would be 32/3 because parts below the axis count negatively.

Q7. Find the total area enclosed between the curve y = x³ − 2x² − x + 2 and the x-axis.

• A. 16/3
• B. 9/4
• C. 25/12
• D. 37/12(correct answer)

Answer: D.The x-intercepts are −1, 1, 2. Integrate |f| over each interval between consecutive intercepts and add: 37/12 square units; the signed integral over the whole span would be 9/4 because parts below the axis count negatively.

Q8. Find the total area enclosed between the curve y = x³ − 6x² + 8x and the x-axis.

• A. 9
• B. 8(correct answer)
• C. 0
• D. 4

Answer: B.The x-intercepts are 0, 2, 4. Integrate |f| over each interval between consecutive intercepts and add: 8 square units; the signed integral over the whole span would be 0 because parts below the axis count negatively.

Q9. Find the total area enclosed between the curve y = x³ − 4x and the x-axis.

• A. 16
• B. 7
• C. 8(correct answer)
• D. −4

Answer: C.The x-intercepts are −2, 0, 2. Integrate |f| over each interval between consecutive intercepts and add: 8 square units; the signed integral over the whole span would be 0 because parts below the axis count negatively.

Q10. Find the total area enclosed between the curve y = x³ − 6x² + 11x − 6 and the x-axis.

• A. 1
• B. 1/4
• C. −9/4
• D. 1/2(correct answer)

Answer: D.The x-intercepts are 1, 2, 3. Integrate |f| over each interval between consecutive intercepts and add: 1/2 square units; the signed integral over the whole span would be 0 because parts below the axis count negatively.