Integrals - ISC Class 12 Mathematics Questions with Answers
105 past-paper questions on Integrals from ISC Class 12 Mathematics papers (2027-2017), newest first. Open one to see its answer.
- On solving $\int -\frac{3}{\sqrt{1-x^2}} dx$, Anil's answer was $\int -\frac{3}{\sqrt{1-x^2}} dx = 3 \cos^{-1} x + c$ and… 2027 · 1 mark · MCQ
- Evaluate the following integrals: Evaluate: $\int \tan x dx$. Hence, evaluate… 2027 · 5 marks · Short answer
- Let $\int_1^5 3f(x) dx = 12$. Show that $\int_5^1 f(x) dx = -4$. Find the value of… 2027 · 3 marks · Short answer
- Shown below is a solved anti-differentiation problem to obtain $f(x)$: $\frac{d}{dx}f(x) = \frac{1}{x(\log x)^2}$ such that… 2027 · 1 mark · MCQ
- Evaluate the following integrals: Evaluate: $\int \ln x dx$. Hence, evaluate: $\int \frac{\ln[\ln(\frac{1+x}{1-x})]}{1-x^2} dx$. 2027 · 5 marks · Short answer
- $\int_{-1}^1 \frac{x^3}{\cos x} dx = 0$ 2027 · 1 mark · Assertion-reason
- Evaluate: $\int \frac{dx}{x^3 + 4x^2 + 4x + 1}$ 2026 · 6 marks · Short answer
- Using properties of definite integral, prove that: $\int_0^\pi \frac{x}{1 + \sin x} \, dx = \pi$ 2026 · 6 marks · Derivation
- Evaluate: $\int \frac{10^x \log 10 + 10x^9}{10^x + x^{10}} \, dx$ 2026 · 1 mark · Short answer
- The value of $\int_{0}^{3/2} |x| \, dx$ is: 2026 · 1 mark · MCQ
- Using properties of definite integral, calculate the value of: $\int_{0}^{\pi/2} \frac{\sin^2 x}{1 + \sin x \cos x} \, dx$ 2026 · 6 marks · Short answer
- Evaluate: $\int \frac{x^2+x+1}{(x+2)(x^2+1)}\,dx$ (Evaluate) 2026 · 6 marks · Short answer
- The value of $\int_{-2}^{1} |x + 1| \, dx$ is: 2026 · 1 mark · MCQ
- Evaluate: $\int_{\pi/4}^{3\pi/4} \frac{x}{1+\sin x}\,dx$ (Evaluate) 2026 · 6 marks · Short answer
- If $\int x^5 \cos(x^6)\,dx = k \sin(x^6) + C$, find the value of ‘$k$’. (Evaluate) 2026 · 2 marks · Numerical
- Evaluate: $\int \frac{e^x}{1 + e^{2x}} \, dx$ 2026 · 2 marks · Short answer
- Evaluate: $\int \frac{\sin x \, dx}{\cos x(1 - \sin x)}$ 2026 · 6 marks · Short answer
- Evaluate: $\int_{0}^{5} \frac{\sqrt{x}}{\sqrt{5-x} + \sqrt{x}}\,dx$ (Evaluate) 2026 · 2 marks · Numerical
- For any function $f(x)$, we have… 2025 · 4 marks · Short answer
- $\int \frac{\cos(\log x)}{x}\,dx$ is equal to: 2025 · 1 mark · MCQ
- Evaluate: $\int \frac{\cos x}{3\cos x - 5}\,dx$ 2025 · 4 marks · Short answer
- Evaluate: $\int \frac{dx}{x^{1/2} - x^{1/3}}$. 2025 · 2 marks · Short answer
- Evaluate: $\int \frac{dx}{\sqrt[4]{(x-1)^3 (x+2)^5}}$. 2025 · 4 marks · Short answer
- Evaluate: $\int_0^{\pi/2} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}}\,dx$ 2025 · 6 marks · Short answer
- Evaluate: $\int \{f(ax+b)\}^n \cdot f'(ax+b) \, dx, n \neq -1$. 2025 · 2 marks · Short answer
- Evaluate: $\int 2^{2^{2^x}} \cdot 2^{2^x} \cdot 2^x \, dx$. 2025 · 2 marks · Short answer
- If $\int \left(\frac{2-x}{(x-1)^2}\right) e^x \, dx = e^x f(x) + c$, then $f(x)$ will be: 2025 · 1 mark · MCQ
- Evaluate: $\int \frac{5}{\sqrt{2x+7}}\,dx$ 2025 · 1 mark · Short answer
- Evaluate: $\int 2^x [f'(x) + f(x)\log 2] \, dx$. 2025 · 2 marks · Short answer
- Identify the correct answer of $\int_0^\pi \sin^{2024} x \cos^{2023} x \, dx$: 2025 · 1 mark · MCQ
- Evaluate: $\int \frac{2x^3 - 1}{x^4 + x} \\, dx$ [Evaluate] 2025 · 2 marks · Short answer
- Evaluate: $\int \frac{\cos x}{\sin^2 x + 4\sin x + 5}\,dx$ 2025 · 4 marks · Short answer
- Evaluate: $\int_{-\pi/2}^{\pi/2} \sin|x| \, dx$. 2025 · 1 mark · Short answer
- Evaluate: $\int \frac{1}{\operatorname{cosec} x + \cot x}\,dx, x \neq 0$ 2025 · 6 marks · Short answer
- Evaluate: $\int \frac{dx}{(x-1)^2(x^2 + 1)^2}$ 2025 · 6 marks · Short answer
- If $\int x^3 5^{x^2}\,dx = K 5^{x^2} + C$, then find the value of $K$. 2025 · 2 marks · Numerical
- Prove: $\int_0^{\frac{3\pi}{4}} \frac{x\,dx}{1 + \sin x} = (\sqrt{2} - 1)\pi$ 2025 · 6 marks · Derivation
- Evaluate: $\int x \tan^{-1} x\,dx$ 2025 · 4 marks · Short answer
- Evaluate: $\int e^x \csc x (1 - \cot x) \\, dx$ [Evaluate] 2025 · 2 marks · Short answer
- $\int_0^1 \frac{x^4 - 1}{x^2 + 1}\,dx$ is equal to: 2025 · 1 mark · MCQ
- Evaluate: $\int \sqrt{\sec\frac{x}{2} - 1} \\, dx$ [Application] 2025 · 4 marks · Short answer
- Evaluate: $\int e^x \left(\frac{1+x}{(2+x)^2}\right)\,dx$ 2025 · 2 marks · Short answer
- Evaluate $\int_{-a}^a x f(x)\,dx$ given, $f(x) = x + g(x)$ where $g(x)$ is an even function. 2025 · 2 marks · Short answer
- Statement I: $\int_{-2}^2 \frac{x^2}{1 + 2^x} \, dx = \frac{8}{3}$. Statement II… 2025 · 1 mark · MCQ
- Evaluate: $\int \frac{\tan\frac{x}{4}}{1 - \sin\frac{x}{4}} \, dx$. 2025 · 6 marks · Short answer
- Evaluate: $\int (\log x)^2\,dx$ 2025 · 4 marks · Short answer
- Evaluate: $\int_0^\pi (\sin^4 x + \cos^4 x) \\, dx$ Hence evaluate… 2025 · 6 marks · Short answer
- Evaluate the following integral: $\int \left(\frac{\cot x + \cot^3 x}{1 + \cot^3 x}\right) \, dx$. 2025 · 6 marks · Short answer
- Evaluate: $\int_0^{\sqrt{2}} [x^2] \\, dx$ [Evaluate] 2025 · 2 marks · Short answer
- Evaluate: $\int_0^{\pi/2} \log\tan x\,dx$ 2025 · 2 marks · Short answer
- If $\int (\cot x - \operatorname{cosec}^2 x)e^x dx = e^x f(x) + c$ then $f(x)$ will be: 2024 · 1 mark · MCQ
- The value of $\int_1^{\sqrt{3}} \frac{dx}{1 + x^2}$ is: 2024 · 1 mark · MCQ
- Evaluate: $\int x^2 \cos x dx$ 2024 · 4 marks · Short answer
- Evaluate: $\int \frac{x + 7}{x^2 + 4x + 7} dx$ 2024 · 4 marks · Short answer
- Evaluate: $\int_0^6 |x + 3| dx$ 2024 · 1 mark · Short answer
- Evaluate: $\int_0^{2\pi} \frac{1}{1 + e^{\sin x}} dx$ 2024 · 2 marks · Short answer
- Evaluate: $\int_0^1 |2x + 1|\,dx$ 2023 · 1 mark · Numerical
- Evaluate: $\int_{-1}^1 x^{17}\cos^4 x\,dx$ 2023 · 1 mark · MCQ
- Evaluate: $\int_0^{\frac{\pi}{2}} \frac{a \sin x + b \cos x}{\sin x + \cos x} \\, dx$ 2023 · 2 marks · Short answer
- Evaluate: $\int \frac{1 + \cos x}{\sin^2 x}\,dx$ 2023 · 1 mark · Short answer
- Evaluate: $\int \frac{2}{(1 - x)(1 + x^2)}\,dx$ 2023 · 6 marks · Short answer
- Evaluate: $\int \frac{1}{\sin^4 x + \cos^4 x} \\, dx$ 2023 · 6 marks · Long answer
- Evaluate: $\int_{-1}^2 |x^3 - x| \\, dx$ 2023 · 4 marks · Short answer
- Evaluate: $\int \cos^{-1}(\sin x)\,dx$ 2023 · 2 marks · Short answer
- Evaluate: $\int \frac{x}{x^2 + 1}\,dx$ 2023 · 1 mark · MCQ
- Evaluate: $\int [\sin(\log x) + \cos(\log x)] \\, dx$ 2023 · 2 marks · Short answer
- If $\int \log 2x \\, dx = x \log 2x - k + c$ where $k$ is a function of $x$, then find $k$. 2023 · 1 mark · Short answer
- Evaluate: $\int \sqrt{\csc x - 1} \\, dx$ 2023 · 4 marks · Short answer
- If $\int x^5 \cos(x^6)\,dx = k \sin(x^6) + C$, find the value of $k$. 2023 · 2 marks · Short answer
- Evaluate: $\int \frac{3e^{2x} - 2e^x}{e^{2x} + 2e^x - 8}\,dx$ 2023 · 6 marks · Short answer
- Evaluate: $\int \log_{10} x \, dx$ 2022 · 2 marks · Short answer
- Evaluate: $\int_{2}^{8} \frac{\sqrt{10-x}}{\sqrt{x} + \sqrt{10-x}} \, dx$ 2022 · 2 marks · Short answer
- Evaluate: $\int \frac{x^3 - x^2 + x - 1}{x - 1} \, dx$ 2022 · 2 marks · Short answer
- Evaluate: $\int \frac{dx}{\sin x + \sin 2x}$ 2022 · 6 marks · Short answer
- $\int_{0}^{2a} f(x) \, dx = \int_{0}^{a} f(x) \, dx + \int_{0}^{a} f(k-x) \, dx$, then the value of $k$ is: 2022 · 1 mark · MCQ
- If $\int \frac{(\log x)^2}{x} \, dx = \frac{(\log x)^k}{k} + c$, then the value of $k$ is: 2022 · 1 mark · MCQ
- Evaluate: $\int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx$ 2022 · 6 marks · Short answer
- Evaluate: $\int \frac{1 + \sin x}{1 - \sin x} \, dx$ 2022 · 4 marks · Short answer
- Given $\int e^x \left(\frac{x-1}{x^2}\right) \, dx = e^x f(x) + c$. Then $f(x)$ satisfying the equation is: 2022 · 1 mark · MCQ
- Evaluate: $\int_{-6}^3 |x + 3| dx$ 2021 · 4 marks · Short answer
- Evaluate: $\int_{-\pi/2}^{\pi/2} \frac{\cos x}{1 + e^x} dx$ 2021 · 6 marks · Short answer
- Evaluate: $\int \tan^{-1}\sqrt{x} dx$ 2021 · 6 marks · Short answer
- Evaluate: $\int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx$ 2021 · 2 marks · Short answer
- Evaluate: $\int_0^\pi \frac{x \tan x}{\sec x + \tan x} dx$ 2021 · 4 marks · Short answer
- Evaluate: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x} dx$ 2021 · 2 marks · Short answer
- Evaluate: $\int \tan^{-1}\sqrt{\frac{1-x}{1+x}} \, dx$. 2020 · 6 marks · Short answer
- Evaluate: $\int \frac{dx}{\sqrt{5x - 4x^2}}$. 2020 · 4 marks · Short answer
- Evaluate: $\int \sin^3 x \cos^4 x \, dx$. 2020 · 4 marks · Short answer
- Evaluate: $\int \frac{2x + 7}{x^2 - x - 2} \, dx$. 2020 · 6 marks · Short answer
- Evaluate: $\int_{4}^{5} |x - 5| \, dx$. 2020 · 2 marks · Short answer
- Evaluate: $\int_0^\pi \frac{x\tan x}{\sec x + \tan x}\\,dx$ 2019 · 6 marks · Short answer
- Evaluate: $\int_{-6}^3 |x + 3|\\,dx$ 2019 · 4 marks · Short answer
- Evaluate: $\int \frac{x(1 + x^2)}{1 + x^4}\\,dx$ 2019 · 4 marks · Short answer
- Evaluate: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x}\\,dx$ 2019 · 2 marks · Short answer
- Evaluate: $\int \frac{6x+7}{\sqrt{(x-5)(x-4)}} \, dx$ 2018 · 4 marks · Short answer
- Evaluate: $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x} \, dx$ 2018 · 2 marks · Short answer
- Evaluate: $\int \frac{3x+5}{x^3 - x^2 - x + 1} \, dx$ 2018 · 6 marks · Long answer
- Evaluate: $\int \frac{x-1}{\sqrt{x^2-x}} \, dx$. 2018 · 6 marks · Short answer
- Evaluate: $\int \frac{x^3 + 5x^2 + 4x + 1}{x^2} \, dx$. 2018 · 2 marks · Short answer
- Evaluate: $\int_{0}^{\pi/2} \frac{\cos^2 x}{1 + \sin x \cos x} \, dx$. 2018 · 6 marks · Short answer
- Evaluate: $\int_{1}^{3} (x^2 + x) \, dx$, expressing as a limit of sum. 2018 · 4 marks · Short answer
- Evaluate: $\int \tan^{-1}\sqrt{x} \, dx$. 2018 · 4 marks · Short answer
- Evaluate: $\int \frac{1}{x^2}\sin^2\left(\frac{1}{x}\right) dx$ 2017 · 3 marks · Short answer
- Evaluate: $\int_{0}^{\pi/4} \log(1 + \tan\theta) \, d\theta$ 2017 · 3 marks · Short answer
- Evaluate: $\int \frac{\sin 2x}{(1 + \sin x)(2 + \sin x)} \, dx$ 2017 · 5 marks · Short answer
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