Integrals in past papers.

CBSE Board Examination 2026

• Board 2026 Q8The constant $A$ in $\int \frac{3ax}{b^2 + c^2x^2}\,dx$
  Integration by substitution1 mark
• Board 2026 Q9$\int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1}\,dx$
  Definite integrals: odd function property1 mark
• Board 2026 Q26Evaluate $\int_0^1 x\tan^{-1}x\,dx$
  Definite integral by parts3 marks
• Board 2026 Q27Find $\int\sqrt{\frac{x + 2}{x - 2}}\,dx$⁠, or $\int\frac{x^2}{(x^2 + 9)(x^2 + 16)}\,dx$
  Integrals by rationalising; partial fractions3 marks
• Board 2026 Q28Show that $I_1 - 4I_2 = 0$ for two definite integrals
  Definite integrals: even-function property3 marks

CBSE Board Examination 2025

• Board 2025 Q3Value of $\int_{-1}^{1} \frac{|x|}{x}\,dx$
  Definite integral of |x|/x1 mark
• Board 2025 Q10Find k if $\int \frac{2^{1/x}}{x^2}\,dx = k \cdot 2^{1/x} + C$
  Integration by substitution1 mark
• Board 2025 Q28$\int\frac{x + \sin x}{1 + \cos x}\,dx$⁠, or $\int_0^{\pi/4}\frac{dx}{\cos^3x\sqrt{2\sin 2x}}$
  Integration by parts; definite by substitution3 marks

CBSE Board Examination 2024

• Board 2024 Q6$\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$
  Definite integrals: properties1 mark
• Board 2024 Q23Find $\int x\sqrt{1 + 2x}\,dx$⁠, or evaluate $\int_0^{\pi^2/4}\frac{\sin\sqrt x}{\sqrt x}\,dx$
  Integration by substitution2 marks
• Board 2024 Q28Find $\int\frac{x^2}{(x^2 + 4)(x^2 + 9)}\,dx$⁠, or evaluate $\int_1^3(|x - 1| + |x - 2| + |x - 3|)\,dx$
  Partial fractions; definite integral of |x − a|3 marks
• Board 2024 Q33Evaluate $\int_0^{\pi/4}\frac{\sin x + \cos x}{9 + 16\sin 2x}\,dx$⁠, or $\int_0^{\pi/2}\sin 2x\tan^{-1}(\sin x)\,dx$
  Definite integrals by substitution and by parts5 marks

CBSE Board Examination 2023

• Board 2023 Q5$f(x)$ when $\frac{d}{dx}(f(x)) = \log x$
  Integration by parts1 mark
• Board 2023 Q6$\int_0^{\pi/6} \sec^2\left(x - \frac{\pi}{6}\right)dx$
  Definite integral of a standard form1 mark
• Board 2023 Q26$\int_0^{\pi/2}[\log(\sin x) - \log(2\cos x)]\,dx$
  Definite integrals: the f(a − x) property3 marks
• Board 2023 Q27$\int\frac{dx}{\sqrt x(\sqrt x + 1)(\sqrt x + 2)}$
  Substitution, then partial fractions3 marks
• Board 2023 Q31$\int_0^{\pi/2}e^x\sin x\,dx$⁠, or $\int\frac{dx}{\cos(x - a)\cos(x - b)}$
  Integration by parts; trig identity integral3 marks

CBSE Board Examination 2022 (Term 2)

• Board 2022 T2 Q3$F(x)$ from $F'(x) = \frac{\sec^4x}{\operatorname{cosec}^4x}$⁠, or $\int \frac{\log x - 3}{(\log x)^4}\,dx$
  Integrals: tan⁴x; the eᵗ[f(t) + f′(t)] form2 marks
• Board 2022 T2 Q9$\int \frac{dx}{\sqrt x + \sqrt[3]x}$⁠, or $\int_0^{\pi/2} \frac{\cos x}{(1 + \sin x)(4 + \sin x)}\,dx$
  Integration by substitution; partial fractions3 marks
• Board 2022 T2 Q13$\int_0^{\pi} \frac{x}{1 + \sin x}\,dx$
  Definite integrals: the property f(a − x)4 marks

CBSE Sample Paper 2026-27

• Sample 2026-27 Q12Find $P$ in $\int \frac{1}{1 + e^{2x}}\,dx = x - P\log(1 + e^{2x}) + C$
  Integration by substitution1 mark
• Sample 2026-27 Q19$\int_{-1}^{1}|x - 2|\,dx = 4$ and the even-function property
  Definite integrals: properties (even functions)1 mark
• Sample 2026-27 Q21Evaluate $\int \frac{dx}{x\sqrt{x^6 - 1}}$⁠, or $\int \frac{\cos x - \sin x}{\sqrt{\sin 2x}}\,dx$
  Integration by substitution2 marks
• Sample 2026-27 Q33Evaluate $\int_{-a}^{a}\sqrt{\frac{a - x}{a + x}}\,dx$
  Definite integrals: even and odd functions5 marks

CBSE Sample Paper 2025-26

• Sample 2025-26 Q11$\int_a^b x\,f(x)\,dx$ when $f(a + b - x) = f(x)$
  Definite integrals: the f(a + b − x) property1 mark
• Sample 2025-26 Q12$a$ in $\int x^3\sin^4(x^4)\cos(x^4)\,dx = a\sin^5(x^4) + C$
  Integration by substitution1 mark
• Sample 2025-26 Q23Find $\int\frac{(x - 3)}{(x - 1)^3}e^x\,dx$⁠, or the shaded area beside $x^2 = y$
  Integral of eˣ(f + f′); area beside a parabola2 marks
• Sample 2025-26 Q33Evaluate $\int_0^1\frac{\log(1 + x)}{1 + x^2}\,dx$⁠, or find $\int\frac{(3\sin\theta - 2)\cos\theta}{5 - \cos^2\theta - 4\sin\theta}\,d\theta$
  Definite integral by property; partial fractions5 marks

CBSE Sample Paper 2024-25

• Sample 2024-25 Q12$\int\frac{dx}{x^3(1 + x^4)^{1/2}}$
  Integration by substitution1 mark
• Sample 2024-25 Q13$\int_0^{2\pi}\operatorname{cosec}^7x\,dx$
  Definite integrals: the f(2a − x) property1 mark
• Sample 2024-25 Q29$\int\left\{\frac{1}{\log x} - \frac{1}{(\log x)^2}\right\}dx$⁠, or $\int_0^1 x(1 - x)^n\,dx$
  Integration by parts; definite integral property3 marks

CBSE Sample Paper 2023-24

• Sample 2023-24 Q9$\int_{-\pi}^{\pi} e^{\cos^2x}\sin^3(2n + 1)x\,dx$
  Definite integrals: odd function property1 mark
• Sample 2023-24 Q24Evaluate $\int_{-1}^{1}\log\left(\frac{2 - x}{2 + x}\right)dx$
  Definite integrals: odd function property2 marks
• Sample 2023-24 Q26Find $\int\frac{2x^2 + 3}{x^2(x^2 + 9)}\,dx$
  Integration by partial fractions (put x² = t)3 marks
• Sample 2023-24 Q28Find $\int\sqrt{\frac{x}{1 - x^3}}\,dx$⁠, or evaluate $\int_0^{\frac{\pi}{4}}\log(1 + \tan x)\,dx$
  Integration by substitution; f(a − x) property3 marks

CBSE Sample Paper 2022-23

• Sample 2022-23 Q5$f(x)$ when $f'(x) = x + \frac{1}{x}$
  Antiderivative from a given derivative1 mark
• Sample 2022-23 Q9$\int_2^3 \frac{x}{x^2 + 1}\,dx$
  Definite integral by substitution1 mark
• Sample 2022-23 Q26Find $\int\frac{dx}{\sqrt{3 - 2x - x^2}}$
  ∫ dx/√(a² − x²) after completing the square3 marks
• Sample 2022-23 Q28$\int_{\pi/6}^{\pi/3}\frac{dx}{1 + \sqrt{\tan x}}$⁠, or $\int_0^4|x - 1|\,dx$
  Definite integrals: properties; modulus integrand3 marks
• Sample 2022-23 Q31Find $\int\frac{x^3 + x + 1}{x^2 - 1}\,dx$
  Partial fractions after long division3 marks

CBSE Sample Paper 2021-22 (Term 2)

• Sample 2021-22 T2 Q1$\int \frac{\log x}{(1 + \log x)^2}\,dx$⁠, or $\int \frac{\sin 2x}{\sqrt{9 - \cos^4x}}\,dx$
  Integration by parts; substitution2 marks
• Sample 2021-22 T2 Q7$\int \frac{x + 1}{(x^2 + 1)x}\,dx$
  Partial fractions with a quadratic factor3 marks
• Sample 2021-22 T2 Q11$\int_{-1}^{2} |x^3 - 3x^2 + 2x|\,dx$
  Definite integral of a modulus function4 marks

CBSE Sample Paper 2020-21

• Sample 2020-21 Q7$\int e^x(1 - \cot x + \text{cosec}^2x)\,dx$⁠. Or: $\int_{-\pi/2}^{\pi/2} x^2\sin x\,dx$
  ∫eˣ(f(x) + f′(x))dx; integral of an odd function1 mark
• Sample 2020-21 Q23$\int \frac{dx}{\cos^2 x(1 - \tan x)^2}$⁠, or $\int_0^1 x(1 - x)^n\,dx$
  Substitution; property of definite integrals2 marks
• Sample 2020-21 Q33$\int \frac{x^2 + 1}{(x^2 + 2)(x^2 + 3)}\,dx$
  Partial fractions in x²3 marks

Questions that also use Integrals

Their main topic is in another chapter.

• Board 2026 Q10Area bounded by $y = x|x|$⁠, the $x$-axis, $x = -1$ and $x = 1$
  Area bounded by a curve and the x-axis1 mark
• Board 2026 Q38Case study: a roundabout and its pond, areas by integration
  Area of a circle by integration4 marks
• Board 2025 Q17Area bounded by $y^2 = x$⁠, $x = 4$ and the x-axis
  Area under y² = x by integration1 mark
• Board 2025 Q31Sketch $y = |x + 3|$ and find the area between $x = -6$ and $x = 0$
  Area under y = |x + 3| by integration3 marks
• Board 2024 Q34Area of the ellipse $\frac{x^2}{16} + \frac{y^2}{4} = 1$ between $x = -2$ and $x = 2$
  Area of part of an ellipse between two lines5 marks
• Board 2023 Q35Area bounded by $x^2 = y$⁠, $y = x + 2$ and the $x$-axis
  Area bounded by a parabola, a line and the x-axis5 marks
• Board 2022 T2 Q11Area of $\{4x^2 + 9y^2 \le 36,\ 2x + 3y \ge 6\}$⁠, or under $x - y + 1 = 0$ from $x = -2$ to $x = 3$
  Area: ellipse and a line; under a line4 marks
• Sample 2026-27 Q34Area between the circle $y = \sqrt{4 - x^2}$ and $x + y = 2$⁠; or between $y^2 = 2x$ and $x - 2y = 0$
  Area between a curve and a line5 marks
• Sample 2025-26 Q28$\int_{-4}^{2}|x + 1|\,dx$ on the graph, or the area between $x^2 = 4y$ and $y = x$
  Area by integration: |x + 1|, parabola and line3 marks
• Sample 2024-25 Q14General solution of $e^{y'} = x$
  Differential equations: variables separable1 mark
• Sample 2024-25 Q18Area bounded by $x^2 = 4y$ and the line $y = 4$
  Area between a parabola and a line1 mark
• Sample 2024-25 Q32Area bounded by $y = 20\cos 2x$⁠, $x = \frac{\pi}{6}$⁠, $x = \frac{\pi}{3}$ and the x-axis
  Area between y = 20 cos 2x and the x-axis5 marks
• Sample 2023-24 Q32Area of $\{(x, y): 0 \le y \le x^2 + 1, 0 \le y \le x + 1, 0 \le x \le 2\}$
  Area of a region bounded by a parabola and a line5 marks
• Sample 2022-23 Q32Area of $\{(x, y): 0 \le y \le x^2, 0 \le y \le x, 0 \le x \le 2\}$
  Area of a region bounded by a parabola and lines5 marks
• Sample 2021-22 T2 Q12Area enclosed by $x + y = 2$⁠, $y^2 = x$ and the $x$-axis, or of $\{0 \le y \le \sqrt3x,\ x^2 + y^2 \le 4\}$
  Area: parabola and a line; circle and a line4 marks
• Sample 2020-21 Q8Area bounded by $y = x^2$⁠, the x-axis, $x = -1$ and $x = 1$
  Area under a parabola1 mark
• Sample 2020-21 Q24Area bounded by $y^2 = 8x$ and the line $x = 2$
  Area bounded by a parabola and a line2 marks
• Sample 2020-21 Q34Area between $x^2 + y^2 = 4$⁠, $y = \sqrt3x$ and the x-axis, or the area of $x^2 + 9y^2 = 36$
  Area of a circle sector; area of an ellipse3 marks