∫ x³ dx is x⁴ ÷ 4 + c and ∫ (2x + 1) dx is x² + x + c, so this Year 12 sheet opens with the power rule and its first constant term. The key also records a definite value of 1 and a final expanded antiderivative.
Key Takeaways
∫ x³ dx = x⁴ ÷ 4 + c (Q1).
∫ (2x + 1) dx = x² + x + c (Q2).
Q5 records 2 as a definite integral value.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.∫ x³ dx =
(a) x⁴ ÷ 4 + c(b) 3x² + c(c) x² + c(d) x⁴ + c
2.∫ (2x + 1) dx =
(a) x² + x + c(b) 2x² + x + c(c) x² + 1 + c(d) 2x + c
3.The value of ∫₀¹ 2x dx is:
(a) 1(b) 2(c) 0(d) ½
4.∫ 1 dx =
(a) x + c(b) 0(c) 1 + c(d) x² + c
5.The area enclosed between y = x, the x-axis, x = 0 and x = 2 is:
(a) 2(b) 4(c) 1(d) 3
6.∫ (3x² + 4x − 1) dx =
(a) x³ + 2x² − x + c(b) 3x³ + 2x² − x + c(c) x³ + 4x − 1 + c(d) 6x + 4 + c
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Find ∫ (x² + 3) dx.
8.Find ∫ (6x² − 2x) dx.
9.Evaluate ∫₀² x² dx.
10.Find ∫ 1 ÷ x² dx.
11.Evaluate ∫₁³ 2x dx.
12.Given f′(x) = 3x² and f(1) = 2, find f(x).
13.Find the area enclosed between y = x², the x-axis, x = 0 and x = 3.
14.Evaluate ∫₀¹ (3x² + 2x) dx.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A car travels with velocity v = 2t + 3 m/s. Find the distance travelled between t = 1 and t = 4 seconds.
16.Water flows into a tank at r = 3t² litres per minute. Find the total amount of water added between t = 0 and t = 2 minutes.
17.Find the area of the region enclosed by the curve y = 2x − x² and the x-axis.
18.A curve has gradient dy/dx = 2x + 1 and passes through the point (1, 4). Find the equation of the curve.
19.A particle moves with velocity v = 4 − 2t m/s. Find the displacement of the particle between t = 0 and t = 2.
20.Find the exact area under the curve y = x³ between x = 0 and x = 2.
Answer Key
1. a) x⁴ ÷ 4 + c
2. a) x² + x + c
3. a) 1
4. a) x + c
5. a) 2
6. a) x³ + 2x² − x + c
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is ∫ x³ dx?
x⁴ ÷ 4 + c, option (a) in this worksheet (Q1).
What is ∫ (2x + 1) dx?
x² + x + c, option (a) in this worksheet (Q2).
What is ∫ x dx?
x² ÷ 2 + c, recorded as option (a) in this worksheet.
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