Embibe Experts Solutions for Chapter: Definite Integration, Exercise 3: Exercise-3

The value of ${\int }_{\sqrt{\ln 2}}^{\sqrt{\ln 3}}\frac{x\sin x^2}{\sin x^2+\sin \left(\ln 6-x^2\right)}dx$ is

For $a\in R$ (the set of all real numbers), $a\ne -1,\underset{n\to \infty }{\lim }\frac{\left(1^a+2^a+\dots .+n^a\right)}{{\left(n+1\right)}^{a-1}\left[\left(na+1\right)+\left(na+2\right)+\dots +\left(na+n\right)\right]}=\frac{1}{60}$. Then $a=$

Let $f:\left(0,\infty \right)\to R$ be given by $f\left(x\right)=\underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t}$. Then 

Let $f:R\to R$ be a continuous odd function, which vanishes exactly at one point and $f\left(1\right)=\frac{1}{2}$. Suppose that $F\left(x\right)={\int }_{-1}^{x}f\left(t\right)dt$ for all $x\in \left[-1,2\right]$ and $G\left(x\right)={\int }_{-1}^{x}t\left|f\left(f\left(t\right)\right)\right|dt$ for all $x\in \left[-1,2\right]$. If $\underset{x\to 1}{\lim }\frac{F\left(x\right)}{G\left(x\right)}=\frac{1}{14}$, then the value of $f\left(\frac{1}{2}\right)$ is

If $g\left(x\right)={\int }_{\sin x}^{\sin \left(2x\right)}{\sin }^{-1}\left(t\right)dt$, then

For each positive integer $n$, let $y_n=\frac{1}{n}{\left(\left(n+1\right)\left(n+2\right)\left(n+3\right).....\left(n+n\right)\right)}^{\frac{1}{n}}$
For $x\in R$, let $\left[·\right]$ be the greatest integer less than or equal to $x$. If $\underset{n\to \infty }{\lim }{y}_n=L$, then the value of $\left[L\right]$ is

The value of the integral ${\int }_0^{\frac{1}{2}}\frac{1+\sqrt{3}}{{\left((x+1{)}^2(1-x{)}^6\right)}^{\frac{1}{4}}}dx$ is

The value of ${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{dx}{\left[x\right]+\left[\sin x\right]+4},$ where $\left[t\right]$ denotes the greatest integer less than or equal to $t,$ is