∫ x dx is x² ÷ 2 + c and ∫ 3x² dx is x³ + c, so this IB sheet opens with the two most used power integrals. The key adds sin x + c, ln|x| + c and eˣ + c as the standard results with 1/2 recorded as a definite value.
Key Takeaways
∫ x dx = x² ÷ 2 + c (Q1).
∫ 3x² dx = x³ + c (Q2).
Q3 records sin x + c as an integral result.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.What is ∫ x dx?
(a) x² ÷ 2 + c(b) x + c(c) x² + c(d) 1 + c
2.What is ∫ 3x² dx?
(a) x³ + c(b) 3x³ + c(c) 6x + c(d) x³ ÷ 3 + c
3.What is ∫ cos x dx?
(a) sin x + c(b) −sin x + c(c) tan x + c(d) cos x + c
4.What is the value of ∫ from 0 to 1 of x dx?
(a) 1(b) 1/2(c) 2(d) 0
5.What is ∫ 1/x dx?
(a) ln|x| + c(b) 1 ÷ x² + c(c) x + c(d) eˣ + c
6.What is ∫ eˣ dx?
(a) eˣ + c(b) xeˣ + c(c) ln x + c(d) 1 + c
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Find ∫ (4x³ + 1) dx.
8.Find ∫ 6x⁵ dx.
9.Evaluate the definite integral from 1 to 2 of 2x dx.
10.Find ∫ sin x dx.
11.Find ∫ (2x + 3) dx.
12.Evaluate the definite integral from 0 to π of sin x dx.
13.Find the area under the curve y = x² from x = 0 to x = 2.
14.If dy/dx = 2x and y = 5 when x = 0, find y in terms of x.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A car's velocity is v(t) = 2t + 1 m/s. Find the distance travelled from t = 1 to t = 3.
16.Find the area of the region bounded by y = x and the x-axis from x = 0 to x = 3.
17.Marginal cost is MC = 3 + 2x dollars per unit. Find the total cost increase from x = 0 to x = 5.
18.A particle's velocity is v(t) = 3t². Find the displacement from t = 0 to t = 2.
19.Water flows into a tank at 2t m³ per hour. Find the total volume of water from t = 0 to t = 4.
20.If dP/dt = 100 × e to the 0.5t and P(0) = 0, find P(2) to the nearest whole number.
Answer Key
1. a) x² ÷ 2 + c
2. a) x³ + c
3. a) sin x + c
4. b) 1/2
5. a) ln|x| + c
6. a) eˣ + 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 ∫ x dx?
x² ÷ 2 + c, option (a) in this worksheet (Q1).
What is ∫ 3x² dx?
x³ + c, option (a) in this worksheet (Q2).
What is ∫ cos x dx?
sin x + c, option (a) in this worksheet (Q3).
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