Definite Integration

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Showing 388-392 of 392 questions.

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JEE Main 2019 (Online) 8th April Evening ShiftQuestion 388

Let $f(x)=\int_0^x g(t)\,dt$, where $g$ is a non-zero even function. If $f(x+5)=g(x)$, then $\int_0^x f(t)\,dt$ equals

Definite Integration

$\int_5^{x+5}g(t)\,dt$$\int_{x+5}^{5}g(t)\,dt$$5\int_{x+5}^{5}g(t)\,dt$$2\int_5^{x+5}g(t)\,dt$

JEE Main 2023 (Online) 30th January Morning ShiftQuestion 389

If $[t]$ denotes the greatest integer $\le t$, then the value of $\frac{3(e-1)}{e}\int_1^2x^2e^{[x]+[x^3]}\,dx$ is :

Definite Integration

$e^9-e$$e^8-e$$e^7-1$$e^8-1$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 390

If $f:\mathbf{R}\to\mathbf{R}$ is given by $f(x)=x+1$, then the value of $\displaystyle\lim_{n\to\infty}\dfrac{1}{n}\left[f(0)+f\left(\dfrac{5}{n}\right)+f\left(\dfrac{10}{n}\right)+\cdots+f\left(\dfrac{5(n-1)}{n}\right)\right]$ is :

Definite Integration

$\dfrac{3}{2}$$\dfrac{5}{2}$$\dfrac{1}{2}$$\dfrac{7}{2}$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 391

Let $g(t)=\displaystyle\int_{-\pi/2}^{\pi/2}\cos\left(\dfrac{\pi}{4}t+f(x)\right)dx$, where $f(x)=\log_e\left(x+\sqrt{x^2+1}\right)$, $x\in\mathbf{R}$. Then which one of the following is correct?

Definite Integration

$g(1)=g(0)$$\sqrt{2}\,g(1)=g(0)$$g(1)=\sqrt{2}\,g(0)$$g(1)+g(0)=0$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 392

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}\left[[x]-\sin x\right]dx$ is equal to :

Definite Integration

$-\pi$$\pi$$0$$1$