∫ sin x dx is −cos x + c and ∫ cos x dx is sin x + c, so this Year 13 sheet opens with the two trigonometric integrals. ∫eˣ dx and ∫1/x dx give eˣ + c and ln|x| + c in the answers, with 2 and e − 1 as definite results.
Key Takeaways
∫ sin x dx = −cos x + c (Q1).
∫ cos x dx = sin 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.∫ sin x dx =
(a) −cos x + c(b) cos x + c(c) sin x + c(d) −sin x + c
2.∫ cos x dx =
(a) sin x + c(b) −sin x + c(c) cos x + c(d) −cos x + c
3.∫ eˣ dx =
(a) eˣ + c(b) x eˣ + c(c) −eˣ + c(d) eˣ
4.∫ 1 ÷ x dx =
(a) ln|x| + c(b) x ln x + c(c) 1 ÷ x² + c(d) x + c
5.The value of ∫₀ to π sin x dx is:
(a) 2(b) 0(c) 1(d) −2
6.The value of ∫₀¹ eˣ dx is:
(a) e − 1(b) e(c) 1(d) e + 1
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Find ∫ (2x + 1)³ dx.
8.Find ∫ 2x eˣ² dx.
9.Find ∫ 1 ÷ (2x + 1) dx.
10.Use the substitution u = x² + 1 to evaluate ∫₀¹ 2x (x² + 1)³ dx.
11.Use integration by parts to find ∫ x cos x dx.
12.Evaluate ∫₀ to π ÷ 2 cos x dx.
13.The curve y = x² is rotated through 360° about the x-axis between x = 0 and x = 2. Find the volume of revolution.
14.Find ∫ 2x ÷ (x² + 1) dx.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Find the area of the region bounded by the curve y = sin x and the x-axis from x = 0 to x = π.
16.A particle moves with velocity v = cos t m/s. Find its displacement between t = 0 and t = π ÷ 2.
17.A solid of revolution is formed by rotating the curve y = x³ through 360° about the x-axis from x = 0 to x = 1. Find the volume.
18.The rate of growth of a population is dP ÷ dt = 1 ÷ t thousand per year. Find the total growth between t = 1 and t = e.
19.Use the substitution u = x² + 1 to evaluate ∫₀¹ 2x (x² + 1)² dx.
20.A bowl is modelled by rotating the curve y = √x through 360° about the x-axis from x = 0 to x = 4. Find the volume of the bowl.
Answer Key
1. a) −cos x + c
2. a) sin x + c
3. a) eˣ + c
4. a) ln|x| + c
5. a) 2
6. a) e − 1
7. Refer to solution guide.
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End of Worksheet
People Also Ask
What is ∫ sin x dx?
−cos x + c, option (a) in this worksheet (Q1).
What is ∫ cos x dx?
sin x + c, option (a) in this worksheet (Q2).
What is ∫ 1/x dx?
ln|x| + c, option (a) in this worksheet (Q4).
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