∫ x² dx gives x³/3 + C and ∫ cos x dx gives sin x + C, so this Class 12 sheet opens with the two basic integration results. The key adds eˣ + C for an exponential integral and 1/2 for a definite value.
Key Takeaways
∫ x² dx = x³/3 + C (Q1).
∫ cos x dx = sin x + C (Q2).
Q4 records eˣ + C as an integral result.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.∫ x² dx = ?
(a) x³/3 + C(b) x³ + C(c) 2x + C(d) x²/2 + C
2.∫ cos x dx = ?
(a) -sin x + C(b) sin x + C(c) tan x + C(d) -cos x + C
3.∫ 1 dx = ?
(a) x + C(b) 0(c) 1 + C(d) x² + C
4.∫ eˣ dx = ?
(a) eˣ + C(b) x·eˣ + C(c) 1/x + C(d) eˣ/2 + C
5.The value of the definite integral ∫₀¹ x dx is:
(a) 1(b) 1/2(c) 0(d) 2
6.∫ (2x + 1) dx = ?
(a) x² + x + C(b) 2x² + x + C(c) x² + 1 + C(d) 2 + C
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the antiderivative of sin x with respect to x.
8.Find ∫ (3x² + 2x) dx.
9.Evaluate the definite integral ∫₀¹ 2x dx.
10.Find ∫ (1/x) dx for x > 0.
11.Evaluate the definite integral of 3 dx from 0 to 2.
12.Find ∫ √x dx.
13.Evaluate the definite integral of sin x from 0 to π.
14.If F(x) is an antiderivative of f(x), write ∫ f(x) dx in terms of F(x).
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.The marginal cost of producing x units is 3x² + 10 and the fixed cost is ₹500. Find the total cost function C(x).
16.The marginal revenue of a firm is given by 20 - 2x. Find the total revenue function, given that revenue is zero when no units are sold.
17.A particle moves with velocity v(t) = 2t + 1 m/s. Find the distance covered by the particle from t = 0 to t = 3 seconds.
18.The rate of growth of a population is dP/dt = 200t. Find the increase in the population from t = 0 to t = 3 years.
19.A body moves with velocity v(t) = t² m/s. Find the distance travelled from t = 1 to t = 2 seconds.
20.The rate of change of temperature of a body is dT/dt = -2t. If the temperature at t = 0 is 40°C, find T(t) and the temperature at t = 3.
Answer Key
1. a) x³/3 + C
2. b) sin x + C
3. a) x + C
4. a) eˣ + C
5. b) 1/2
6. a) x² + x + C
7. Refer to solution guide.
8. Refer to solution guide.
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20. Refer to solution guide.
End of Worksheet
People Also Ask
What is ∫ x² dx?
x³/3 + C, option (a) in this worksheet (Q1).
What is ∫ cos x dx?
sin x + C, option (b) in this worksheet (Q2).
What is ∫ eˣ dx?
eˣ + C, option (a) in this worksheet (Q4).
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