Q JEE MAIN 2026

Let $f$ be a differentiable function satisfying $f(x)=1-2 x+\int_0^x \mathrm{e}^{(x-t)} f(t) \mathrm{dt}, x \in \mathbf{R}$ and let $...

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Q JEE MAIN 2026

The value of $\sum_{\mathrm{r}=1}^{20}\left(\left|\sqrt{\pi\left(\int_0^{\mathrm{r}} \mathrm{x}|\sin \pi \mathrm{x}| \mathrm{dx}\right)}\right|\right)$ is $\_\_\_\_$ .

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Q JEE MAIN 2026

Let $f$ be a polynomial function such that $f\left(\mathrm{x}^2+1\right)=\mathrm{x}^4+5 \mathrm{x}^2+2$, for all $\mathrm{x} \in \mathbb{R}$. Then $\int_0^3 f(\mathrm{x}) \mathrm{dx}$ is...

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Q JEE MAIN 2026

Let [.] denote the greatest integer function. Then $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right) \mathrm{d} x$ is equal to :

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Q JEE MAIN 2026

The number of elements in the set $\mathrm{S}=\left\{x: x \in[0,100]\right.$ and $\left.\int_0^x t^2 \sin (x-t) d t=x^2\right\}$ is $\_\_\_\_$ .

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Q JEE MAIN_2026_

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(\mathrm{t}))^2+\left(f^{\prime}(\mathrm{t})\right)^2\right) d t$. Then the mean of $...

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Q JEE MAIN 2026

Let $[\cdot]$ be the greatest integer function. If $\alpha=\int_0^{64}\left(x^{1 / 3}-\left[x^{1 / 3}\right]\right) \mathrm{d} x$, then $...

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Q JEE MAIN_2026_

The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5 \pi}{24}} \frac{\mathrm{~d} x}{1+\sqrt[3]{\tan 2 x}}$ is :

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Q JEE MAIN 2026

Let $f(x)=[x]^2-[x+3]-3, x \in \mathbf{R}$, where $[\cdot]$ is the greatest integer function. Then Options

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Q JEE MAIN_2026_

The value of $\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

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Q JEE MAIN_2026_

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int_1^x f(t) d t=3 x f(x)+x^3-4$ for all $...

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Q JEE MAIN 2026

If $\int_0^1 4 \cot ^{-1}\left(1-2 x+4 x^2\right) \mathrm{d} x=\operatorname{atan}^{-1}(2)-\operatorname{blog}_{\mathrm{e}}(5)$, where $\mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $(2 \mathrm{a}+\mathrm{b})$ is equal to

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Q JEE MAIN 2026

Let $[\cdot]$ denote the greatest integer function and $f(x)=\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}^3} \sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{\mathrm{k}^2}{3^x}\right]$. Then $12 \sum_{\mathrm{j}=1}^{\infty} f(\mathrm{j})$ is equal...

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Q JEE MAIN 2026

$6 \int_0^\pi|(\sin 3 x+\sin 2 x+\sin x)| d x$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2026

The value of $\int_{-\pi / 6}^{\pi / 6}\left(\frac{\pi+4 x^{11}}{1-\sin (|x|+\pi / 6)}\right) d x$ is equal to:

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Q JEE MAIN_2019_

$\lim _{n \rightarrow \infty}\left(\frac{(n+1)^{1 / 3}}{n^{4 / 3}}+\frac{(n+2)^{1 / 3}}{n^{1 / 3}}+\ldots . .+\frac{(2 n)^{1 / 3}}{n^{4 / 3}}\right)$ is...

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Q JEE MAIN 2019

The value of $\int_0^{\pi / 2} \frac{\sin ^3 x}{\sin x+\cos x} d x$ is :

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Q JEE MAIN_2019_

The value of $\int_0^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$, where $[t]$ denotes the greatest integer function, is:

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Q JEE MAIN 2019

A value of $\alpha$ such that $$ \int_\alpha^{\alpha+1} \frac{\mathrm{dx}}{(\mathrm{x}+\alpha)(\mathrm{x}+\alpha+1)}=\log _{\mathrm{e}}\left(\frac{9}{8}\right) \text { is: } $$

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Q JEE MAIN 2020

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of $...

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Q JEE Main 2020

The integral $\int_0^2 \| x-1|-x| d x$ is equal to $\_\_\_\_$ .

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Q JEE-MAIN 2019

Let $f$ and $g$ be continuous functions on $[0$, a $]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $...

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Q JEE MAIN 2019

$ \int_0^{\pi / 3} \frac{\tan \theta}{\sqrt{2 \mathrm{ksec} \theta}} \mathrm{~d} \theta=1-\frac{1}{\sqrt{2}},(\mathrm{k}>0) $ then the value of $k$ is

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Q JEE MAIN 2019

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)| \leq 2|x-y|^{\frac{3}{2}}$, for all $...

JEE Main Mathematics Hard

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Q JEE MAIN 2020

If for all real triplets $(a, b, c), f(x)=a+b x+c x^2$; then $\int_0^1 f(x) d x$ is equal to

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Q JEE MAIN 2020

The value of $\int_0^{2 \pi} \frac{x \sin ^8 x}{\sin ^8 x+\cos ^8 x} d x$ is equal to

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Q JEE MAIN 2020

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+\mathrm{e}^{\sin x}} \mathrm{dx}$ is:

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Q JEE MAIN 2019

$\lim _{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots \ldots+\frac{1}{5 n}\right)$ is equal to :

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Q JEE MAIN 2019

The integral $\int_1^e\left\{\left(\frac{x}{e}\right)^{2 x}-\left(\frac{e}{x}\right)^x\right\} \log _e x d x$ is equal to

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Q JEE MAIN 2020

If $I_1=\int_0^1\left(1-x^{50}\right)^{100} d x$ and $I_2=\int_0^1\left(1-x^{50}\right)^{101} d x$ such that $I_2=\alpha I_1$ then $\alpha$ equals to:

JEE Main Mathematics Hard

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Q JEE MAIN 2020

$$ \text { The integral } \int_1^2 e^x \cdot x^x\left(2+\log _e x\right) d x \text { equals : } $$

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Q JEE MAIN 2019

$\int_0^{\pi / 3} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta=1-\frac{1}{\sqrt{2}},(k>0)$, then the value of $k$ is

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Q JEE MAIN 2019

The value of the integral $\int_{-2}^2 \frac{\sin ^2 x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}$ (where $[x]$ denotes the greatest integer less than or equal to...

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Q JEE MAIN 2019

The integral $\int_{\pi / 6}^{\pi / 3} \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3} x d x$ is equal...

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Q JEE MAIN_2021_

The value of the inkegral $\int_0^1 \frac{\sqrt{x} d x}{(1+x)(1+3 x)(3+x)}$ is :

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Q JEE MAIN 2020

The integral $$ \int_{\pi / 6}^{\pi / 3} \tan ^3 x \cdot \sin ^2 3 x\left(2 \sec ^2 x \cdot...

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Q JEE MAIN 2020

Let a function $f:[0,5] \rightarrow R$ be continuous, $f(1)=3$ and $F$ be defined as : $$ F(x)=\int_1^x t^2 g(t) d...

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Q JEE Main 2019

Let $I=\int_a^b\left(x^4-2 x^2\right) d x$ If $I$ is minimum then the ordered pair $(a, b)$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2020

$\lim _{x \rightarrow 0} \frac{\int_0^x t \sin (10 t) d t}{x}$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2019

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)| \leqslant 2|x-y|^{\frac{3}{2}}$, for all $...

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Q JEE MAIN 2019

If $f: R \rightarrow R$ is a differentiable function and $f(2)=6$, then $...

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Q JEE MAIN 2019

The area (in sq. units) of the region $A=\left\{(x, y): \frac{y^2}{2} \leq x \leq y+4\right\}$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2019

The value of the integral $\int_0^1 x \cot ^{-1}\left(1-x^2+x^4\right) d x$ is

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Q JEE-MAIN 2020

If the value of the integral $\int_0^{1 / 2} \frac{x^2}{\left(1-x^2\right)^{3 / 2}} d x$ is $\frac{k}{6}$, then $k$ is equal...

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Q JEE MAIN 2021

If $x \phi(\mathrm{x})=\int_5^x\left(3 t^2-2 \phi^{\prime}(\mathrm{t})\right) d t, x>-2$, and $\phi(0)=4$, then $\phi(2)$ is

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $[\mathrm{t}]$ denote the greatest integer $\leq 1$. Then the value of 8 . $\int_{-\frac{1}{2}}^1([2 x]|\mathrm{x}|) d x$ is

JEE Main Mathematics Easy

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Q JEE Main 2021

If $[x]$ is the greatest integer $\leq x$, then $\pi^2 \int_0^2\left(\sin \frac{\pi x}{2}\right)(x-[x]) \quad d x$ is equal to :

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Q JEE MAIN 2021

If $\frac{d y}{d x}=\frac{2^{x+y}-2^x}{2 y}, \mathrm{y}(0)=1$, then $\mathrm{y}(1)$ is equal to :

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Q JEE MAIN 2019

If $\int_0^x f(t) d t=x^2+\int t^2 f(t) d t$ then $f(1 / 2)$ is :

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Q JEE MAIN 2019

The value of $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{d x}{[x]+[\sin x]+4}$ where $[t]$ denotes the greatest integer less than or equal to $t$, is...

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Q JEE Main 2021

If $\frac{d y}{d x}=\frac{2^x y+2^y \cdot 2^x}{2^x+2^{x+y} \log _e 2}, y(0)$, then for $y=1$, the value of $x$ lies in...

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Q JEE MAIN 2021

Let f be a non - negative function in $[0,1]$ and twice differentiable in $(0,1)$. If $...

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Q JEE MAIN 2021

If $\int_6^{16} \frac{\log _e x^2}{\log _e x^2+\log _e\left(x^2-44 x+484\right)} d x$ is equal to :

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Q JEE MAIN 2021

If $U_n=\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)^2 \cdots\left(1+\frac{n^2}{n^2}\right)^n$, then $\lim _{n \rightarrow \infty}\left(U_n\right)^{n^{\frac{-4}{2}}}$ is equal to :

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Q JEE Main 2021

The value of $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2 n-1} \frac{n^2}{n^2 4 r^2}$ is:

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Q JEE Main 2021

The value of $\int_{\frac{-1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}\left(\left(\frac{x+1}{x-1}\right)^2+\left(\frac{x-1}{x+1}\right)^2-2\right)^{\frac{1}{2}} d x$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If $f(x)=\frac{2-x \cos x}{2+x \cos x}$ and $g(x)=\log _e x,(x>0)$ then the value of the integral $...

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Q JEE MAIN 2021

The value of $\int_{-\frac{\pi}{2}}\left(\frac{\sin \pi^{\sin x}}{1+\pi^2}\right) \mathrm{dx}$

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Q JEE Main 2019

The value of $\int_0^\pi|\cos x|^3 d x$ is :

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Q JEE MAIN 2021

If the value of the integral $\int_0^5 \frac{x+[x]}{e^{x-[x]}} d x=\alpha e^{-1}+\beta$, where $a, b \in R, 5 \alpha+6 \beta=0$, and...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

The function $f(x)$, that satisfies the condition $f(x)=x+\int_0^{\pi / 2} \sin x \cdot \cos y f(y) d y$, is :

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Q JEE MAIN 2021

Let $J_{n, m}=\int_0^{\frac{1}{2}} \frac{x^n}{x^m-1} d x, \forall n>m$ and $m \in N$. Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ where...

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Q JEE MAIN 2021 S2

Let $f: R \rightarrow R$ be a continuous function. Then $...

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Q JEE MAIN 2021 S2

If $\int_0^\pi\left(\sin ^3 x\right) e^{-\sin ^2 x} d x=\alpha-\frac{\beta}{e} \int_0^1 \sqrt{t} e^t d t$, then $\alpha+\beta$ is equal to $\_\_\_\_$...

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Q JEE MAIN_2021

Let $f:[0, \infty) \rightarrow[0, \infty)$ be defined as $f:(x)=\int_0^x[y] d y$ where $[x]$ is the greatest integer less than or...

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Q JEE MAIN_2021_

The value of the definite integral $\int_{\pi / 24}^{5 \pi / 24} \frac{d x}{1+\sqrt[3]{\tan 2 x}}$ is :

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Q JEE MAIN 2019

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

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If $I_{m, n}=\int_0^1 x^{m-1}(1-x)^{n-1} d x$, for $m, n \geq 1$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $f(x)=\int_0^x e^t f(t) d t+e^x$ be a differentiable function for all $x \in R$. Then $f(x)$ equals :

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let the domain of the function $f(x)=\log _4\left(\log _5\left(\log _3\left(18 x-x^2-77\right)\right)\right)$ be $(a, b)$. Then the value of the integral...

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

If $f(x)=\left\{\begin{array}{cc}\int_0^x(5+|1-t|) d t, & x>2 \\ 5 x+1, & x \leq 2\end{array}\right.$, then

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The value of the definite integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{\left(1+e^{x \cos x}\right)\left(\sin ^4 x+\cos ^4 x\right)}$ is equal to :

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The value of $\lim _{x \rightarrow \infty} \frac{1}{n} \sum_{j=1}^n \frac{(2 j-1)+8 n}{(2 j-1)+4 n}$ is equal to :

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Q JEE MAIN 2021

The value of $\int_{-2}^2\left|3 x^2-3 x-6\right| d x$ is $\_\_\_\_$ .

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Q _JEE MAIN_2021_

If $\int_0^{100 \pi} \frac{\sin ^2 x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}} d x=\frac{\alpha \pi^3}{1+4 \pi^2}, \alpha \in R$ where $[x]$ is the greatest integer less...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

The value of the integral $\int_{-1}^1 \log _e(\sqrt{1-x}+\sqrt{1+x}) d x$ is equal to :

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let a be a positive real number such that $\int_0^a e^{x-[x]} d x=10 e-9$ where $[x]$ is the greatest integer...

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Q JEE MAIN 2021

$\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^2}+\frac{n}{(n+2)^2}+\ldots . .+\frac{n}{(2 n-1)^2}\right]$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $P(x)$ be real polynomial of degree 3 which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local...

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Q JEE MAIN 2021

$$ \text { If } I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^n x   d x, \text { then : } $$

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Q JEE MAIN 2021

Let $g(x)=\int_0^x f(t) d t$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3} \leq f(t) \leq 1$ for...

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Q JEE-MAIN 2021

If the integral $\int_0^{10} \frac{[\sin 2 \pi x]}{e^{x-[x]}} d x=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma$, where, $\alpha, \beta, \gamma$ are integers and $[x]$...

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Q JEE-MAIN 2021

Let $f: R \rightarrow R$ be defined as $f(x)=e^{-x} \sin x$. If $F:[0,1] \rightarrow R$ is a differentiable function such...

JEE Main Mathematics Hard

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Q JEE MAIN_2021_

If [ ] represents the greatest integer function, then the value of

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

Which of the following statements is incorrect for the function $\mathrm{g}(\mathrm{a})$ for $\alpha \in \mathrm{R}$ such that

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Q JEE MAIN 2021

Let $g(t)=\int_{-\pi / 2}^{\pi / 2}\left(\cos \frac{\pi}{4} t+f(x)\right) d x$, where $f(x)=\log _e\left(x+\sqrt{x^2+1}\right), x \in R$. Then which one of...

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Q JEE MAIN 2021

Let $\mathrm{P}(\mathrm{x})=\mathrm{x}^2+\mathrm{bx}+\mathrm{c}$ be a quadratic polynomial with real coefficients such that $=\int_0^1 P(x) d x=1$ and $\mathrm{P}(\mathrm{x})$ leaves remainder 5...

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Q JEE MAIN 2021

If $f: R \rightarrow R$ is given by $f(x)=x+1$, then the value of $\lim _{n \rightarrow \infty}\left[f(x)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\ldots+f\left(\frac{5(n-1)}{n}\right)\right]$, is :

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Q JEE MAIN 2021

If $\int_{-a}^a(|x|+|x-2|) d x=22(a>2)$ and $[x]$ denotes the greatest integer $\leq x$, then $\int_a^{-a}(x+[x]) d x$ is equal to $\_\_\_\_$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

$\lim _{x \rightarrow 0} \frac{\int_0^{x^2}(\sin \sqrt{t}) d t}{x^3}$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Consider the integral $$ \int_0^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x $$ where $[x]$ denotes the greatest integer less than or equal...

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Q JEE MAIN 2021

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $f(x)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime}(x)=f^{\prime}(2-x)$ for all $x \in(0,2), f(0)=1$ and $f(2)=e^2$. Then...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The value of the integral, $\int_1^3\left[x^2-2 x-2\right] d x$, where $[x]$ denotes the greatest integer less than or equal to...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $f(x)$ and $g(x)$ be two functions satisfying $f\left(x^2\right)+ g(4-x)=4 x^3$ and $g(4-x)+g(x)=0$, then the value of $...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

If the normal to the curve $y(x)=\int_0^x\left(2 t^2-15 t+10\right) d t$ at a point $(a, b)$ is parallel to the...

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Q JEE-MAIN 2021

Let $f:(0,2) \rightarrow \mathbb{R}$ be defined as

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Q JEE MAIN 2021

The value of the integral $\int_0^\pi|\sin 2 x| d x$ is

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^2 x}{1+3 x} d x$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2021

For $x>0$, if $f(x)=\int_1^x \frac{\log _e t}{(1+t)} d t$ then $f(e)+f\left(\frac{1}{e}\right)$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The value of $\sum_{n=1}^{100} \int_{n-1}^n e^{x-[x]} d x$ where $[x]$ is the greatest integer $\leq \mathrm{X}$, is

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The value of $\int_{-1}^1 x^2 e^{\left[x^3\right]} d x$ where $[t]$ denotes the greatest integer $\leq t$, is:

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Q JEE MAIN_2022

If $n(2 n+1) \int_0^1\left(1-x^n\right)^{2 n} d x=1177 \int_0^1\left(1-x^n\right)^{2 n+1} d x, n \in N$ is equal to $\_\_\_\_$

JEE Main Mathematics Medium

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Q JEE MAIN_2022

If $a=\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{2 n}{n^2+k^2}$ and $f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1)$, then :

JEE Main Mathematics Hard

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Q JEE MAIN 2022

If $\lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)]=33 . \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}}\left[1^k+2^k+3^k+\ldots . .+n^k\right]$, then the integral...

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Q JEE MAIN 2022

For any real number $x$, let $[x]$ denote the largest integer less than equal to $x$. Let $f$ be a...

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Q JEE MAIN 2022

The integral $\frac{24}{\pi} \int_0^{\sqrt{2}} \frac{\left(2-x^2\right) d x}{\left(2+x^2\right) \sqrt{4+x^4}}$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If $\int_0^2\left(\sqrt{2 x}-\sqrt{2 x-x^2}\right) d x=\int_0^1\left(1-\sqrt{1-y^2}-\frac{y^2}{2}\right) d y+\int_1^2\left(2-\frac{y^2}{2}\right) d y+1$

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $f$ be a real valued continuous function on $[0,1]$ and $f(x)=x+\int_0^1(x-t) f(t) d t$. Then which of the following...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be continuous function satisfying $f(x)+f(x+k)=n$, for all $x \in R$ where $k>0$ and $n$ is...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a differentiable function such that $f\left(\frac{\pi}{4}\right)=\sqrt{2}, f\left(\frac{\pi}{2}\right)=0$ and $f\left(\frac{\pi}{2}\right)=1$ and let $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The value of $b>3$ for which $12 \int_3^b \frac{1}{\left(x^2-1\right)\left(x^2-4\right)} d x=\log _e\left(\frac{49}{40}\right)$, is equal to

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Q JEE MAIN 2022

The integral $\int_0^1 \frac{1}{7^{\left[\frac{1}{x}\right]}} \mathrm{dx}$, where [.] denotes the greatest integer function is equal to :

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f$ be a differentiable function in $\left(0, \frac{\pi}{2}\right)$. If $\int_{\cos x}^1 t^2 f(t) d t=\sin ^3 x+\cos x$ then...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

$\int_0^5 \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x$, Where $[t]$ denotes greatest integer less than or equal to $t$, is equal to :

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The value of the integral $\int_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^x\right)\left(\sin ^6 x+\cos ^6 x\right)}$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If $b_n=\int_0^{\frac{\pi}{2}} \frac{\cos ^2 n x}{\sin x} d x, n \in N$, then

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a function defined by: $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The value of the integral $\frac{48}{\pi^4} \int_0^\pi\left(\frac{3 \pi x^2}{2}-x^3\right) \frac{\sin x}{1+\cos ^2 x} d x$ is equal to

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let the solution curve $y=y(x)$ of the differential equation $\left(4+x^2\right) d y-2 x\left(x^2+3 y+4\right) d x=0$ pass through the origin....

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f(x)=\max \{|x+1|,|x+2|, \ldots,|x+5|\}$. Then $\int_{-6}^0 f(x) d x$ is equal to $\_\_\_\_$ .

JEE Main Mathematics Easy

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Q JEE MAIN 2022

The value of the integral $\int_{-2}^2 \frac{\left|x^3+x\right|}{\left(e^{x|x|}+1\right)} d x$ is equal to:

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\operatorname{Max}_{0 \leq x \leq 2}\left\{\frac{9-x^2}{5-x}\right\}=\alpha$ and $\operatorname{Max}_{0 \leq x \leq 2}\left\{\frac{9-x^2}{5-x}\right\}=\beta$ If $...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f(\theta)=\sin \theta+\int_{-\pi / 2}^{\pi / 2}(\sin \theta+t \cos \theta) f(t) d t$. Then the value of $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The value of $\int_0^\pi \frac{e^{\cos x} \sin x}{\left(1+\cos ^2 x\right)\left(e^{\cos x}+e^{-\cos x}\right)} d x$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $[t]$ denote the greatest integer function. If $\int_0^{2.4}\left[X^2\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$, then $\alpha+\beta+\gamma+\delta$ is equal to $\_\_\_\_$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

The value of $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $f(x)$ be a positive function and $I_1=\int_{-\frac{1}{2}}^1 2 x f(2 x(1-2 x)) d x$ and $I_2=\int_{-1}^2 f(x(1-x)) d x$....

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^2, \forall x \in R$. Then $\int^\pi f(x) \sin x d x$ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

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

JEE Main Mathematics Hard

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Q JEE MAIN 2023

Let $f$ be a continuous function satisfying $\int_0^{t^2}\left(f(x)+x^2\right) d x=\frac{4}{3} t^3, \forall t>0$. Then $f\left(\frac{\pi^2}{4}\right)$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

$4 \int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right) d x-3 \log _e(\sqrt{3})$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\quad f(x)+2 f\left(\frac{1}{x}\right)=x^2+5 \quad$ and $\quad 2 g(x)-3 g\left(\frac{1}{2}\right)=x, x>0$. If $\quad \alpha=\int_1^2 f(x) \mathrm{d} x, \quad$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function. If $10 \int_1^x f(t) d t=5 x f(x)-x^5-9$ for all $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $(a, b)$ be the point of intersection of the curve $x^2=2 y$ and the straight line $y-2 x-6=0$ in...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2023

$\int_0^1 \frac{1}{\left(5+2 x-2 x^2\right)\left(1+e^{(2-4 x)}\right)} d x=\frac{1}{\alpha} \log _e\left(\frac{\alpha+1}{\beta}\right), \alpha, \beta>0$, then $\alpha^4-\beta^4$ is equal to:

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Among: $$ \begin{aligned} & \left(S_1\right)_{\operatorname{lin}}: \lim _{n \rightarrow \infty} \frac{1}{n^2}(2+4+6+\ldots \ldots+2 n)=1 \\ & \left(S_2\right)_{\ln }: \lim _{n \rightarrow \infty}...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

If $\int_{-0.15}^{0.15}\left|100 \mathrm{x}^2-1\right| \mathrm{dx}=\frac{\mathrm{k}}{3000}$, then k is equal to $\_\_\_\_$ .

JEE Main Mathematics Medium

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Q JEE MAIN 2023

$\int_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+1}$

JEE Main Mathematics Medium

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Q JEE MAIN 2023

For $m, n>0$, let $\alpha(m, n)=\int_0^2 t^m(1+3 t)^n d t$. If $11 \alpha(10,6)+18 \alpha(11,5)=p(14)^6$, then $p$ is equal to

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x>0$. Then $18 \int_1^2 f(x) d x$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $[t]$ denotes the greatest integer $\leq t$. Then $...

JEE Main Mathematics Medium

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Q JEE-Main_2023

$\int_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^x+6} d x$

JEE Main Mathematics Medium

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Q JEE MAIN 2023

The value of the integral $\int_{-\log _e 2}^{\log _e 2} \mathrm{e}^{\mathrm{x}}\left(\log _{\mathrm{e}}\left(\mathrm{e}^{\mathrm{x}}+\sqrt{1+\mathrm{e}^{2 \mathrm{x}}}\right)\right) \mathrm{dx}$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $I(x)=\int \frac{(x+1)}{x\left(1+x e^x\right)^2} d x, x>0$, If $\lim _{x \rightarrow \infty} I(x)=0$, then $I(1)$ is equal to

JEE Main Mathematics Medium

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Q JEE-Main

Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x>0$. Then $18 \int_1^2 f(x) d x$ is equal to :

JEE Main Mathematics Easy

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Q JEE-Main 2024

Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in N$. Then the value of $\sum_{\mathrm{k}=1}^{10} \frac{1}{7\left(\mathrm{r}_{\mathrm{k}}-1\right)}$ is equal to $\_\_\_\_$

JEE Main Mathematics Hard

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Q JEE MAIN 2024

If $\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} d x=\frac{1}{a} \log _e\left(\frac{a}{3}\right)+\frac{\pi}{b \sqrt{3}}$, where $a, \mathrm{~b} \in \mathrm{~N}$, then $\mathrm{a}+\mathrm{b}$...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

The value of ∫_(-π)^π (2y(1+sin⁡y))/(1+〖cos〗^2⁡y) dy is :

JEE Main Mathematics Easy

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Q JEE-Main 2024

For α,β∈R and a natural number n, let

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The integral $\int_0^{\frac{\pi}{4}} \frac{136 \sin x}{3 \sin x+5 \cos x} d x$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $$ \begin{aligned} & f(x)=\frac{4^x}{4^x+2} \text { and } \\ &...

JEE Main Mathematics Easy

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Q JEE-Main 2024

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x d x}{\left(1+e^{\sin x}\right)\left(1+\sin ^4 x\right)}=\alpha \pi+\beta \log _e(3+2 \sqrt{2})$, where...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The value 9∫_0^9 [√(10x/(x+1))]dx, where [t] denotes the greatest integer less than or equal to t, is

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $S=(-1, \infty)$ and $f: S \rightarrow \mathbb{R}$ be defined as $f(x)=\int_{-1}^x\left(e^t-1\right)^{11}(2 t-1)^5(t-2)^7(t-3)^{12}(2 t-10)^{61} d t$ Let $p=$ Sum of...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If the integral $525 \int_0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\cos ^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$ is equal to $(n \sqrt{2}-64)$,...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let $\mathrm{f}:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$, If the $...

JEE Main Mathematics Medium

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Q JEE MAIN

If the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2023}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$ then the value of a is

JEE Main Mathematics Medium

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Q JEE MAIN 2024

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2 x-8$

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The value of $\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}$ is:

JEE Main Mathematics Medium

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Q JEE-Main 2024

The value of the integral $\int_0^{\frac{\pi}{4}} \frac{x d x}{\sin ^4(2 x)+\cos ^4(2 x)}$ equals :

JEE Main Physics Easy

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Q JEE MAIN 2024

Let $a$ be the sum of all coefficients in the expansion of $\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If $\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are rational numbers, then $2 \mathrm{a}+3 \mathrm{~b}-4 \mathrm{c}$ is...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $24 \int_{0}^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) \mathrm{d} x=2 \pi+\alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $f(x)=\int_{0}^{x} \mathrm{t}\left(\mathrm{t}^{2}-9 \mathrm{t}+20\right) \mathrm{dt}, 1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(x)=x f(x)$ for all $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_{0}^{x} t f(\mathrm{t}) \mathrm{dt}$....

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The integral $\int_0^\pi {\frac{{(x + 3)\sin x}}{{1 + 3{{\cos }^2}x}}} dx$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The value of $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the domain of the function $f(x) = {\log _2}{\log _4}{\log _6}\left( {3 + 4x - {x^2}} \right)$ be (a,...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let [.] denote the greatest integer function. If $\int_0^{{e^3}} {\left[ {\frac{1}{{{e^{x - 1}}}}} \right]} dx = \alpha - {\log _e}2$,...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

If $\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $f:(0,\infty ) \to {\bf{R}}$ be a twice differentiable function. If for some $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The integral $80\int_0^{\frac{\pi }{4}} {\left( {\frac{{\sin \theta + \cos \theta }}{{9 + 16\sin 2\theta }}} \right)} d\theta $ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let for some function ${\rm{y}} = f(x),\int_0^x t f(t)dt = {x^2}f(x),x > 0$ and $f(2) = 3$. Then $f(6)$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let f be a differentiable function such that $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If $I(m,n) = \int_0^1 {{x^{m - 1}}} {(1 - x)^{n - 1}}dx,m,n > 0$, then $I(9,14) + I(10,13)$ is

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The value of $...

JEE Main Mathematics Easy

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Q

Let \( y = f(x) \) be the solution of the differential equation \[\frac{dy}{dx} = \frac{y - x^2 + 4x}{\sqrt{1...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let for $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $x=x(y)$ be the solution of the differential equation ${{y}^{2}}~\text{d}x+\left( {x-\frac{1}{y}} \right)\text{d}y=0$. If $x(1)=1$, then $x\left( {\frac{1}{2}} \right)$ is :

JEE Main Chemistry Medium

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Q JEE MAIN 2025

Let $f(x)=\int_0^{x^2} \frac{\mathrm{t}^2-8 \mathrm{t}+15}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of $f$,...

JEE Main Mathematics Easy

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