Integrals — Previous Year Questions (Class 12 Mathematics)
Integrals covers indefinite and definite integration, substitution, partial fractions, and applications (area, volume). CBSE tests fundamental techniques;
TL;DR: Integrals covers indefinite and definite integration, substitution, partial fractions, and applications (area, volume). CBSE tests fundamental techniq…
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Integrals covers indefinite and definite integration, substitution, partial fractions, and applications (area, volume). CBSE tests fundamental techniques;
Integrals — Previous Year Questions with Solutions
Q (2023, 2 marks): Find the integral: ∫(3x^2 + 2x + 1) dx
Answer: ∫(3x^2 + 2x + 1) dx
= ∫3x^2 dx + ∫2x dx + ∫1 dx
= 3 × (x^3/3) + 2 × (x^2/2) + x + C
= x^3 + x^2 + x + C
Final Answer: x^3 + x^2 + x + C
Q (2022, 2 marks): Evaluate the definite integral: ∫[0 to 2] 4x dx
Answer: ∫[0 to 2] 4x dx
= [4 × x^2/2] from 0 to 2
= [2x^2] from 0 to 2
= 2(2)^2 - 2(0)^2
= 2 × 4 - 0
= 8
Final Answer: 8
Q (2021, 3 marks): Find the integral using substitution: ∫ 2x e^(x^2) dx
Answer: Let u = x^2
du = 2x dx
So dx = du / (2x)
∫ 2x e^(x^2) dx = ∫ e^u du
= e^u + C
= e^(x^2) + C
Final Answer: e^(x^2) + C
Q (2023, 3 marks): Integrate by parts: ∫ x e^x dx
Answer: Using integration by parts: ∫ u dv = uv - ∫ v du
Let u = x, dv = e^x dx
Then du = dx, v = e^x
∫ x e^x dx = x × e^x - ∫ e^x dx
= x e^x - e^x + C
= e^x(x - 1) + C
Final Answer: e^x(x - 1) + C
Q (2022, 3 marks): Find the area under the curve y = x^2 between x = 0 and x = 3.
Answer: Area = ∫[0 to 3] x^2 dx
= [x^3/3] from 0 to 3
= (3)^3/3 - (0)^3/3
= 27/3 - 0
= 9 square units
Final Answer: Area = 9 square units
Q (2021, 3 marks): State the fundamental theorem of calculus and explain its two parts.
Answer: Fundamental Theorem of Calculus (two parts):
Part 1: If a function f is continuous on [a, b] and F is an antiderivative of f on [a, b], then:
∫[a to b] f(x) dx = F(b) - F(a)
This connects differentiation and integration (antiderivative).
Part 2: If f is continuous on an open interval containing [a, x], then:
d/dx [∫[a to x] f(t) dt] = f(x)
This shows that the derivative of an integral recovers the original function.
Significance:
1. Allows computation of definite integrals using antiderivatives.
2. Proves that integration and differentiation are inverse operations.
3. Bridges analytical (limits) and computational (algebra) perspectives.
4. Enables calculation of areas under curves and physical quantities like work, displacement.
Example:
∫[0 to 2] 3x^2 dx = [x^3] from 0 to 2 = 8 - 0 = 8 (Part 1)
Final Answer: FTC part 1 enables definite integrals via antiderivatives. Part 2 shows derivative of integral is the original function.
Frequently Asked Questions
What is the difference between indefinite and definite integrals?
Indefinite integral ∫f(x)dx gives a family of antiderivatives F(x) + C. Definite integral ∫[a to b] f(x)dx gives a specific numerical value (area under the curve from a to b).
When should you use integration by parts vs. substitution?
Use substitution when the integrand contains a function and its derivative (e.g., ∫2x e^(x^2)). Use integration by parts when the integrand is a product of two functions that don't fit substitution (e.g., ∫x e^x).
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