Embibe Experts Solutions for Chapter: Definite Integration, Exercise 1: WBJEE 2020

The value of $\sum _{n=1}^{10}{\int }_{-2n-1}^{-2n}{\sin }^{27}xdx+\sum _{n=1}^{10}{\int }_{2n}^{2n+1}{\sin }^{27}xdx$ is equal to

If $x^2+y^2=a^2,$ then ${\int }_0^a\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx=$

The value of $I={\int }_{\pi /2}^{5\pi /2}\frac{e^{{\tan }^{-1}}\left(\sin x\right)}{e^{{\tan }^{-1}\left(\sin x\right)}+e^{{\tan }^{-1}\left(\cos x\right)}}dx,$ is

Let $f$ be a continuous function in $[0,1]$ , then $\underset{n\to \infty }{\lim }\sum _{ j=0}^n\frac{1}{n}f\left(\frac{j}{n}\right)$ is

$\underset{n\to \infty }{\lim }\frac{3}{n}\left\{1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}\right.\left.+\dots +\sqrt{\frac{n}{n+3(n-1)}}\right\}$

The value of $\underset{n\to \infty }{\lim }\frac{1}{n}\left\{{\sec }^2\frac{\pi }{4n}+{\sec }^2\frac{2\pi }{4n}+\dots +{\sec }^2\frac{n\pi }{4n}\right]$ is

Let $I_n={\int }_0^1{x}^n{\tan }^{-1}xdx.$ If $a_n{I}_{n+2}+b_n{I}_n=c_n$ for all $n\ge 1,$ then

Let $I={\int }_0^1\frac{x^3\cos 3x}{2+x^2}dx.$ Then