JEE Mathematics / Practice

Definite Integration

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JEE Main 2021 (Online) 16th March Morning ShiftQuestion 368

Let $f:\mathbf{R}\to\mathbf{R}$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x\in\mathbf{R}$. If $I_1=\displaystyle\int_{0}^{8}f(x)\,dx$ and $I_2=\displaystyle\int_{-1}^{3}f(x)\,dx$, then the value of $I_1+2I_2$ is equal to $\_\_\_\_\_\_\_\_$ .

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JEE Main 2021 (Online) 16th March Morning ShiftQuestion 369

Let $f:(0,2)\to\mathbf{R}$ be defined as $f(x)=\log_2\left(1+\tan\left(\dfrac{\pi x}{4}\right)\right)$. Then, $\displaystyle\lim_{n\to\infty}\dfrac{2}{n}\left(f\left(\dfrac{1}{n}\right)+f\left(\dfrac{2}{n}\right)+\ldots+f(1)\right)$ is equal to $\_\_\_\_\_\_\_\_$ .

Definite Integration

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JEE Main 2024 (Online) 4th April Morning ShiftQuestion 370

Let $f(x)=\begin{cases}-2, & -2\le x\le0\\ x-2, & 0<x\le2\end{cases}$ and $h(x)=f(|x|)+|f(x)|$. Then $\displaystyle\int_{-2}^{2}h(x)\,dx$ is equal to :

Definite Integration

$1$$2$$4$$6$

JEE Main 2024 (Online) 4th April Morning ShiftQuestion 371

If $\displaystyle\int_{0}^{\pi/4}\dfrac{\sin^{2}x}{1+\sin x\cos x}\,dx=\dfrac{1}{a}\log_e\left(\dfrac{a}{3}\right)+\dfrac{\pi}{b\sqrt{3}}$, where $a,b\in\mathbf{N}$, then $a+b$ is equal to $\_\_\_\_\_\_\_\_$ .

Definite Integration

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JEE Main 2024 (Online) 4th April Evening ShiftQuestion 372

If the value of the integral $\displaystyle\int_{-1}^{1}\dfrac{\cos\alpha x}{1+3^{x}}\,dx$ is $\dfrac{2}{\pi}$. Then, a value of $\alpha$ is

Definite Integration

$\dfrac{\pi}{4}$$\dfrac{\pi}{3}$$\dfrac{\pi}{2}$$\dfrac{\pi}{6}$