Definite Integration

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JEE Main 2026 (Online) 8th April Evening ShiftQuestion 21

The value of the integral $\int_{0}^{2}\frac{\sqrt{x\left(x^{2}+x+1\right)}}{\left(\sqrt{x+1}\right)\left(\sqrt{x^{4}+x^{2}+1}\right)}\,dx$ is equal to :

$\frac{1}{3}\log_e\left(3-2\sqrt{2}\right)$$\frac{2}{3}\log_e\left(4+\sqrt{2}\right)$$\frac{2}{3}\log_e\left(3+2\sqrt{2}\right)$$\frac{1}{3}\log_e\left(1+6\sqrt{2}\right)$

JEE Main 2021 (Online) 17th March Evening ShiftQuestion 22

If the integral $\int_{0}^{10}\frac{[\sin2\pi x]}{e^{x-[x]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma$, where $\alpha,\beta,\gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $x$, then the value of $\alpha+\beta+\gamma$ is equal to :

Definite Integration

$0$$10$$20$$25$

JEE Main 2021 (Online) 25th July Evening ShiftQuestion 23

The value of the integral $\int_{-1}^{1}\log\left(x+\sqrt{x^{2}+1}\right)dx$ is :

Definite Integration

$0$$1$$2$$-1$

JEE Main 2025 (Online) 8th April Evening ShiftQuestion 24

Let $f(x)$ be a positive function and $I_1=\int_{-\frac{1}{2}}^{1}2xf(2x(1-2x))\,dx$ and $I_2=\int_{-1}^{2}f(x(1-x))\,dx$. Then the value of $\frac{I_2}{I_1}$ is equal to

Definite Integration

$12$$9$$6$$4$

JEE Main 2024 (Online) 27th January Morning ShiftQuestion 25

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and $I_1=\int_{a}^{b}x\sin\left(4x-x^{2}\right)dx$, $I_2=\int_{a}^{b}\sin\left(4x-x^{2}\right)dx$, then $36\frac{I_1}{I_2}$ is equal to :

Definite Integration

$72$$80$$88$$66$