Definite Integration

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JEE Main 2024 (Online) 27th January Morning ShiftQuestion 56

If $\int_{0}^{1}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=a+b\sqrt{2}+c\sqrt{3}$, where $a,b,c$ are rational numbers, then $2a+3b-4c$ is equal to :

$4$$7$$8$$10$

JEE Main 2025 (Online) 8th April Evening ShiftQuestion 57

The integral $\int_{-1}^{\frac{3}{2}}\left(\left|\pi^{2}x\sin(\pi x)\right|\right)dx$ is equal to :

Definite Integration

$1+3\pi$$2+3\pi$$3+2\pi$$4+\pi$

JEE Main 2025 (Online) 3rd April Evening ShiftQuestion 58

The integral $\int_{0}^{\pi}\frac{8x\,dx}{4\cos^{2}x+\sin^{2}x}$ is equal to

Definite Integration

$\pi^{2}$$\frac{3\pi^{2}}{2}$$2\pi^{2}$$4\pi^{2}$

JEE Main 2023 (Online) 10th April Evening ShiftQuestion 59

Let $f$ be a continuous function satisfying $\int_{0}^{t^{2}}\left(f(x)+x^{2}\right)dx=\frac{4}{3}t^{3}$, $\forall t>0$. Then $f\left(\frac{\pi^{2}}{4}\right)$ is equal to

Definite Integration

$-\pi^{2}\left(1+\frac{\pi^{2}}{16}\right)$$\pi^{2}\left(1-\frac{\pi^{2}}{16}\right)$$-\pi\left(1+\frac{\pi^{3}}{16}\right)$$\pi\left(1-\frac{\pi^{3}}{16}\right)$

JEE Main 2024 (Online) 1st February Evening ShiftQuestion 60

The value of $\int_{0}^{1}\left(2x^{3}-3x^{2}-x+1\right)^{\frac{1}{3}}dx$ is equal to :

Definite Integration

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