$I={\int }_0^{2\mathrm{\pi }}{e}^{{\sin }^{2}x+\sin x+1}dx$, then

Important Questions on Definite Integration

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 114

Let $f^{"}\left(x\right)\ge 0, f^{\text{'}}\left(x\right)>0, f\left(0\right)=3 \& f\left(x\right)$ is defined in $\left[-2, 2\right].$ If $f\left(x\right)$ is non-negative, then

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 115

Let mean value of $f\left(x\right)=\frac{1}{x+c}$ over interval $\left(0,2\right)$ is $\frac{1}{2}\ln 3$, then the positive value of $c$ is

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 116

$\underset{n\to \infty }{\lim } \sum _{r=1}^n\left(\frac{r^3}{r^4+n^4}\right)$ equals to

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 117

$\underset{n\to \infty }{Lt} \sum _{r=2n+1}^{3n}\frac{n}{r^2-n^2}$ is equal to

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 118

$\underset{n\to \infty }{\lim }{\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)\dots \left(1+\frac{n^2}{n^2}\right)\right]}^{\frac{1}{n}}$ is equal to

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 119

$\underset{n\to \infty }{\lim }\frac{\pi }{n}\left\{\sin \frac{\pi }{n}+\sin \frac{2\pi }{n}+\dots +\sin \frac{(n-1)\pi }{n}\right\}$ is equal to

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 120

Let $I_n={\int }_0^1{\left(1-x^3\right)}^{n}dx,\left(n\in N\right)$, then

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-1>Q 121

Let $f:[0,\infty )\to R$ be a continuous and strictly increasing function such that $f^3\left(x\right)={\int }_0^{x}t{f}^2\left(t\right)dt, x\ge 0$. The area enclosed by $y=f\left(x\right)$, the $x$-axis and the ordinate at $x=3$ is -