JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Definite Integrals

More JEE Main questions on Definite Integrals

1. The value of ${\int }_{-\pi /6}^{\pi /6}\left(\frac{\pi +4x^{11}}{1-\sin (|x|+\pi /6)}\right)dx$ is equal to
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2. $6{\int }_0^{\pi }|(\sin 3x+\sin 2x+\sin x)|dx$ is equal to.......
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3. Let $[\cdot ]$ denote the greatest integer function and $f(\mathrm{x})={\lim }_{\mathrm{n}\to \infty }\frac{1}{{\mathrm{n}}^3}{\sum }_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{{\mathrm{k}}^2}{3^{\mathrm{x}}}\right]$. Then $12{\sum }_{\mathrm{j}=1}^{\infty }f(\mathrm{j})$ is equal …
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4. If ${\int }_0^{1}4{\cot }^{-1}\left(1-2\mathrm{x}+4{\mathrm{x}}^2\right)\mathrm{dx}=\mathrm{a}{\tan }^{-1}(2)-\mathrm{b}{\log }_{\mathrm{e}}(5)$, where $\mathrm{a},\mathrm{b}\in \mathbf{N}$, then $(2\mathrm{a}+\mathrm{b})$ is equal to ____.
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5. Let $f:[1,\infty )\to \mathbb{R}$ be a differentiable function. If $6{\int }_1^{x}f(t)dt=3xf(x)+x^3-4$ for all $\mathrm{x}\ge 1$, then the value of $f(2)-f(3)$ is
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6. The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\frac{1}{[\mathrm{x}]+4}\right)\mathrm{dx}$, where [•] denotes the greatest integer function, is
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