If $b_n={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{2}nx}{\sin x}dx,n\in \mathrm{\mathbb{N}}$, then

Important Questions on Definite Integration

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 25 June 2022 Shift 2>Q 110

The value of $b>3$ for which $12{\int }_3^b\frac{1}{\left(x^2-1\right)\left(x^2-4\right)}dx={\log }_e\left(\frac{49}{40}\right)$, is equal to _____. 

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 26 June 2022 Shift 2>Q 111

The integral $\frac{24}{\pi }{\int }_0^{\sqrt{2}}\frac{\left(2-x^2\right)\mathrm{dx}}{\left(2+x^2\right)\sqrt{4+x^4}}$ is equal to ______.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 26 June 2022 Shift 1>Q 112

The value of the integral $\frac{48}{{\pi }^4}{\int }_0^{\pi }\left(\frac{3\pi x^2}{2}-x^3\right)\frac{\sin x}{1+{\cos }^{2}x}dx$ is equal to ______.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 27 June 2022 Shift 1>Q 113

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

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 27 June 2022 Shift 2>Q 114

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f\left(x\right)={\int }_0^{x^2}\frac{t^2-5t+4}{2+e^t}dt$, then the ordered pair $\left(m,n\right)$ is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 27 June 2022 Shift 2>Q 115

Let $f$ be a differentiable function in $\left(0,\frac{\pi }{2}\right)$. If ${\int }_{\cos x}^1{t}^{2}f\left(t\right)dt={\sin }^{3}x+\cos x$, then $\frac{1}{\sqrt{3}}{f}^{\text{'}}\left(\frac{1}{\sqrt{3}}\right)$ is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 27 June 2022 Shift 2>Q 116

The integral ${\int }_0^1\frac{1}{7^{\left[\frac{1}{x}\right]}}dx$, where $\left[·\right]$ denotes the greatest integer function, is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 29 - Definite Integration>JEE Main - 28 June 2022 Shift 1>Q 117

Let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral ${\int }_0^1\left[-8x^2+6x-1\right]dx$ is equal to