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DSE M2 Indefinite Integration Practice

DSE M2 · 60 practice questions with instant marking and worked solutions.

Indefinite Integration is an important part of the DSE M2 mathematics curriculum. This topic develops antiderivatives, integration by substitution, integration by parts, integrals of trigonometric functions and the constant of integration. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Indefinite Integration has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Indefinite Integration; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Indefinite Integration set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Indefinite Integration is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    It is given that $F'(x)=2x-3$ and $F(0)=2$. Find the value of $F(1)$. Choose the correct answer.
  2. 2 Basic
    It is given that $\int(5x-5)^{6}\,dx=A(5x-5)^{7}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Give the answer only.

  3. 3 Basic
    It is given that $\int x(x^{2}+2)^{3}\,dx=A(x^{2}+2)^{4}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Show the essential working and give the answer.

  4. 4 Basic
    It is given that $\int xe^{3x}\,dx=e^{3x}(Ax+B)+C$. The value of the constant $B$ is 1/9. Is this true?
  5. 5 Basic
    It is given that $F'(x)=4x+4$ and $F(0)=6$. Find the value of $F(4)$. Choose the correct answer.
  6. 6 Basic
    It is given that $\int(-2x+5)^{3}\,dx=A(-2x+5)^{4}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Give the answer only.

  7. 7 Basic
    It is given that $\int 4x(x^{2}-5)^{4}\,dx=A(x^{2}-5)^{5}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    It is given that $\int xe^{4x}\,dx=e^{4x}(Ax+B)+C$. The value of the constant $B$ is 1/16. Is this true?
  9. 9 Basic
    It is given that $F'(x)=2x+6$ and $F(0)=1$. Find the value of $F(3)$. Choose the correct answer.
  10. 10 Basic
    It is given that $\int(5x+5)^{4}\,dx=A(5x+5)^{5}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Give the answer only.

  11. 11 Basic
    It is given that $\int 4x(x^{2}-2)^{5}\,dx=A(x^{2}-2)^{6}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Show the essential working and give the answer.

  12. 12 Basic
    It is given that $\int xe^{5x}\,dx=e^{5x}(Ax+B)+C$. The value of the constant $B$ is 1/25. Is this true?
  13. 13 Standard
    It is given that $\int x\cos x\,dx=Ax\sin x+B\cos x+C$. Find the value of the constant $B$. Choose the correct answer.
  14. 14 Standard
    It is given that $\int x^{2}\ln x\,dx=x^{3}(A\ln x+B)+C$ for $x>0$. Find the value of the constant $B$ (as a fraction in lowest terms). Give the answer only.

  15. 15 Standard
    It is given that $\int 2\cos 2x\,dx=B\sin 2x+C$. Find the value of the constant $B$. Show the essential working and give the answer.

  16. 16 Standard DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=9x^{2}-4x-5$, and $\Gamma$ passes through $(-1,9)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=3$.
  17. 17 Standard
    It is given that $\int x^{4}\ln x\,dx=x^{5}(A\ln x+B)+C$ for $x>0$. Find the value of the constant $A$ (as a fraction in lowest terms). Choose the correct answer.
  18. 18 Standard
    It is given that $\int \cos 5x\,dx=B\sin 5x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Give the answer only.

  19. 19 Standard
    It is given that $\int 2\sin 2x\,dx=B\cos 2x+C$. Find the value of the constant $B$. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=-3x^{2}-4x+5$, and $\Gamma$ passes through $(1,-4)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=0$.
  21. 21 Standard
    It is given that $\int x\cos 3x\,dx=Ax\sin 3x+B\cos 3x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Choose the correct answer.
  22. 22 Standard
    It is given that $\int x^{4}\ln x\,dx=x^{5}(A\ln x+B)+C$ for $x>0$. Find the value of the constant $B$ (as a fraction in lowest terms). Give the answer only.

  23. 23 Standard
    It is given that $\int 6\cos 6x\,dx=B\sin 6x+C$. Find the value of the constant $B$. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=6x^{2}+6x-2$, and $\Gamma$ passes through $(2,2)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=-2$.
  25. 25 Advanced
    It is given that $F'(x)=3x-1$ and $F(0)=-3$. Find the value of $F(2)$. Choose the correct answer.
  26. 26 Advanced
    It is given that $\int(3x-3)^{2}\,dx=A(3x-3)^{3}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Give the answer only.

  27. 27 Advanced
    It is given that $\int x(x^{2}+4)^{5}\,dx=A(x^{2}+4)^{6}+C$. The value of the constant $A$ is 1/12. Is this true?
  28. 28 Advanced DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=6x^{2}-8x+4$, and $\Gamma$ passes through $(1,7)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=3$.
  29. 29 Advanced
    It is given that $F'(x)=4x-5$ and $F(0)=-5$. Find the value of $F(3)$. Choose the correct answer.
  30. 30 Advanced
    It is given that $\int(3x+1)^{4}\,dx=A(3x+1)^{5}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Give the answer only.

  31. 31 Advanced
    It is given that $\int xe^{2x}\,dx=e^{2x}(Ax+B)+C$. The value of the constant $B$ is -1/4. Is this true?
  32. 32 Advanced DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=-3x^{2}-4x$, and $\Gamma$ passes through $(-1,-7)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=-2$.
  33. 33 Advanced
    It is given that $F'(x)=4x+2$ and $F(0)=-3$. Find the value of $F(2)$. Choose the correct answer.
  34. 34 Advanced
    It is given that $\int(5x+2)^{6}\,dx=A(5x+2)^{7}+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Give the answer only.

  35. 35 Advanced
    It is given that $\int 3x(x^{2}+3)^{4}\,dx=A(x^{2}+3)^{5}+C$. The value of the constant $A$ is 3/10. Is this true?
  36. 36 Advanced DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=3x^{2}-2x+6$, and $\Gamma$ passes through $(2,4)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=-2$.
  37. 37 Challenging
    It is given that $\int xe^{-2x}\,dx=e^{-2x}(Ax+B)+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Choose the correct answer.
  38. 38 Challenging
    It is given that $\int x\sin 5x\,dx=Ax\cos 5x+B\sin 5x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Show the essential working and give the answer.

  39. 39 Challenging
    It is given that $\int x\ln x\,dx=x^{2}(A\ln x+B)+C$ for $x>0$. The value of the constant $A$ is 1. Is this true?
  40. 40 Challenging DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=9x^{2}-8x-4$, and $\Gamma$ passes through $(2,-8)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=-2$.
  41. 41 Challenging
    It is given that $\int xe^{4x}\,dx=e^{4x}(Ax+B)+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Choose the correct answer.
  42. 42 Challenging
    It is given that $\int x\cos 4x\,dx=Ax\sin 4x+B\cos 4x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Show the essential working and give the answer.

  43. 43 Challenging
    It is given that $\int x\ln x\,dx=x^{2}(A\ln x+B)+C$ for $x>0$. The value of the constant $B$ is 1/4. Is this true?
  44. 44 Challenging DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=3x^{2}-2x-3$, and $\Gamma$ passes through $(-1,1)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=2$.
  45. 45 Challenging
    It is given that $\int xe^{2x}\,dx=e^{2x}(Ax+B)+C$. Find the value of the constant $A$ (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Challenging
    It is given that $\int x\sin 4x\,dx=Ax\cos 4x+B\sin 4x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Show the essential working and give the answer.

  47. 47 Challenging
    It is given that $\int x^{2}\ln x\,dx=x^{3}(A\ln x+B)+C$ for $x>0$. The value of the constant $B$ is 1/9. Is this true?
  48. 48 Challenging DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=-3x^{2}-6x+1$, and $\Gamma$ passes through $(2,-5)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=0$.
  49. 49 Elite
    It is given that $\int \sin 4x\,dx=B\cos 4x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Give the answer only.

  50. 50 Elite
    It is given that $F'(x)=6x-6$ and $F(0)=8$. Find the value of $F(1)$. Show the essential working and give the answer.

  51. 51 Elite
    It is given that $\int(4x-2)^{5}\,dx=A(4x-2)^{6}+C$. The value of the constant $A$ is 1/24. Is this true?
  52. 52 Elite DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=-3x^{2}+8x-5$, and $\Gamma$ passes through $(2,9)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=3$.
  53. 53 Elite
    It is given that $\int 2\sin 6x\,dx=B\cos 6x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Give the answer only.

  54. 54 Elite
    It is given that $F'(x)=4x+5$ and $F(0)=8$. Find the value of $F(3)$. Show the essential working and give the answer.

  55. 55 Elite
    It is given that $\int(4x-1)^{4}\,dx=A(4x-1)^{5}+C$. The value of the constant $A$ is 1/20. Is this true?
  56. 56 Elite DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=-3x^{2}+8x+5$, and $\Gamma$ passes through $(-1,0)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=3$.
  57. 57 Elite
    It is given that $\int 2\sin 3x\,dx=B\cos 3x+C$. Find the value of the constant $B$ (as a fraction in lowest terms). Give the answer only.

  58. 58 Elite
    It is given that $F'(x)=4x-5$ and $F(0)=4$. Find the value of $F(1)$. Show the essential working and give the answer.

  59. 59 Elite
    It is given that $\int(-2x-1)^{4}\,dx=A(-2x-1)^{5}+C$. The value of the constant $A$ is -1/10. Is this true?
  60. 60 Elite DSE long question
    The gradient of curve $\Gamma$ is $\frac{dy}{dx}=3x^{2}$, and $\Gamma$ passes through $(-1,0)$. Answer each part; give the answer only.
    1. (a) Writing the equation of $\Gamma$ as $y=ax^3+bx^2+cx+C$, find $a$.
    2. (b) Find the constant of integration $C$.
    3. (c) Find the $y$-coordinate of $\Gamma$ at $x=2$.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Indefinite Integration

This topic needs particular care with antiderivatives, integration by substitution, integration by parts, integrals of trigonometric functions and the constant of integration. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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DSE M2 mixed revision → My mistakes →

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Repeating the same mistake usually means an earlier concept never landed. We find that step in the trial lesson.

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