JEE Main 2026 (Online) 21st January Evening Shift | Definite Integration Question 29 | Mathematics

If $\int\limits_0^1 4 \cot ^{-1}\left(1-2 x+4 x^2\right) \mathrm{d} x=\mathrm{a\,tan}^{-1}(2)-\mathrm{b\,log}_{\mathrm{e}}(5)$, where $\mathrm{a}, \mathrm{b} \in \mathrm{N}$, then $(2 \mathrm{a}+\mathrm{b})$ is equal to $\_\_\_\_$ .

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JEE Main 2026 (Online) 21st January Morning Shift

Numerical

-1

$6 \int_0^\pi|(\sin 3 x+\sin 2 x+\sin x)| d x$ is equal to $\_\_\_\_$ .

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JEE Main 2025 (Online) 2nd April Morning Shift

Numerical

-1

Let [.] denote the greatest integer function. If $\int_\limits0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$, then $\alpha^3$ is equal to _________.

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JEE Main 2025 (Online) 29th January Evening Shift

Numerical

-1

If $ 24 \int\limits_0^{\frac{\pi}{4}} \bigg[\sin \left| 4x - \frac{\pi}{12} \right| + [2 \sin x] \bigg] dx = 2\pi + \alpha $, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.

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