Definite Integration

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JEE Main 2024 (Online) 8th April Evening ShiftQuestion 46

Let $\int_{\alpha}^{\log_e4}\frac{dx}{\sqrt{e^{x}-1}}=\frac{\pi}{6}$. Then $e^{\alpha}$ and $e^{-\alpha}$ are the roots of the equation :

$x^{2}+2x-8=0$$x^{2}-2x-8=0$$2x^{2}-5x+2=0$$2x^{2}-5x-2=0$

JEE Main 2023 (Online) 6th April Evening ShiftQuestion 47

Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^{2}$, $\forall x\in\mathbb{R}$. Then $\int_{0}^{\pi}f(x)\sin x\,dx$ is equal to

Definite Integration

$2\pi^{2}$$\frac{\pi^{2}}{4}$$\frac{\pi^{2}}{2}$$\pi^{2}$

JEE Main 2024 (Online) 30th January Evening ShiftQuestion 48

Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=ae^{2x}+be^{x}+cx$. If $f(0)=-1$, $f'(\log_e2)=21$ and $\int_{0}^{\log_e4}(f(x)-cx)\,dx=\frac{39}{2}$, then the value of $|a+b+c|$ equals

Definite Integration

$8$$10$$12$$16$

JEE Main 2024 (Online) 8th April Morning ShiftQuestion 49

The value of $k\in\mathbb{N}$ for which the integral $I_n=\int_{0}^{1}\left(1-x^{k}\right)^{n}dx$, $n\in\mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is

Definite Integration

$8$$10$$14$$7$

JEE Main 2023 (Online) 13th April Morning ShiftQuestion 50

$\int_{0}^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^{x}+6}dx=$

Definite Integration

$\log_e\left(\frac{512}{81}\right)$$\log_e\left(\frac{64}{27}\right)$$\log_e\left(\frac{256}{81}\right)$$\log_e\left(\frac{32}{27}\right)$