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JEE Advanced Integral Calculus (Definite Integrals & Differential Equations) Previous Year Questions

~4.2 Qs / Year (Paper 1 & 2)•12.4% Historical Weightage
← Practice JEE Main Foundational PYQs for this chapter
High-Yield Recurring Advanced Concepts
Leibniz Rule for Differentiation under the Integral Sign ($\frac{d}{dx} \int_{u(x)}^{v(x)} f(t, x) dt$)
Definite Integral as the Limit of a Riemann Sum ($\lim_{n \to \infty} \frac{1}{n} \sum f(r/n)$)
Properties of Definite Integrals: King Rule ($\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$)
Linear Differential Equations with Integrating Factors ($IF = e^{\int P dx}$) and Bernoulli Form
Orthogonal Trajectories and Variable Separable Geometric Applications

Chapter PYQs (40 Questions)

Live Partial Marking Mode
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2025 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2025 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2025 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2025 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2024 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2024 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2024 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2024 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2023 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2023 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2023 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2023 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2022 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2022 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2022 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2022 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2021 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2021 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2021 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2021 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2020 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2020 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2020 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2020 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2018 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2018 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2018 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2018 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2017 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2017 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2017 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2017 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2016 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2016 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q42Paper 2MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2016 Paper 2 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q48Paper 2MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2016 Paper 2 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is: