Definite Integration

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JEE Main 2024 (Online) 9th April Evening ShiftQuestion 41

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

Definite Integration

$\sqrt{2}-\sqrt{5}+\log_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$$\sqrt{5}-\sqrt{2}+\log_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$$\sqrt{2}-\sqrt{5}+\log_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$$\sqrt{5}-\sqrt{2}+\log_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$

JEE Main 2023 (Online) 6th April Morning ShiftQuestion 42

Let $5f(x)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3$, $x>0$. Then $18\int_{1}^{2}f(x)\,dx$ is equal to:

Definite Integration

$5\log_e2-3$$5\log_e2+3$$10\log_e2-6$$10\log_e2+6$

JEE Main 2023 (Online) 11th April Evening ShiftQuestion 43

Let the function $f:[0,2]\to\mathbb{R}$ be defined as $f(x)=\begin{cases}e^{\min\{x^{2},x-[x]\}}, & x\in[0,1)\\ e^{[x-\log_ex]}, & x\in[1,2]\end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int_{0}^{2}xf(x)\,dx$ is

Definite Integration

$1+\frac{3e}{2}$$2e-1$$2e-\frac{1}{2}$$(e-1)\left(e^{2}+\frac{1}{2}\right)$

JEE Main 2024 (Online) 30th January Evening ShiftQuestion 44

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\pi/6$ and $\pi/4$, respectively with positive $x$-axis. If $27\int_{1}^{3}\left(\left(f'(t)\right)^{2}+1\right)f''(t)\,dt=\alpha+\beta\sqrt{3}$ where $\alpha,\beta$ are integers, then the value of $\alpha+\beta$ equals

Definite Integration

$26$$36$$-14$$-16$

JEE Main 2025 (Online) 2nd April Evening ShiftQuestion 45

Let $(a,b)$ be the point of intersection of the curve $x^{2}=2y$ and the straight line $y-2x-6=0$ in the second quadrant. Then the integral $I=\int_{a}^{b}\frac{9x^{2}}{1+5^{x}}dx$ is equal to :

Definite Integration

$18$$21$$24$$27$