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

If $n\in I, n\ge 0$, then the value of ${\int }_0^{\pi }\left[\frac{\sin \left(n+{\displaystyle \frac{1}{2}}\right)x}{\sin \left({\displaystyle \frac{x}{2}}\right)}\right]dx$ is

If $a>0$ and $A={\int }_0^a{\cos }^{-1}xdx,$ then ${\int }_{-a}^a\left({\cos }^{-1}x-{\sin }^{-1}\sqrt{1-x^2}\right)dx=\pi a-\lambda A,$ then $\lambda$ is

If $f\left(x\right)=\frac{1}{2^n}$ when $\frac{1}{2^{n+1}}<x\le \frac{1}{2^n}, n=0, 1, 2, \dots ,$ then$\underset{n\to \infty }{\lim }{\int }_{\frac{1}{2^n}}^{1}f\left(x\right)dx$ equals to

If $I={\int }_{-\frac{\pi }{6}}^{\frac{\pi }{6}}\left\{\frac{\pi +4x^5}{1-\sin \left(\left|x\right|+\frac{\pi }{6}\right)}\right\}dx$, then $I$ equals

If ${\int }_0^{\pi /2}\sqrt{\sin 2\theta }.\sin \theta d\theta =\frac{\pi }{n}$, then find $n$

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

The value of $\underset{n\to \infty }{\lim }\frac{(n!{)}^{1/n}}{n}$ is

Let $f$ be a real-valued function defined on the interval $\left(-1, 1\right)$ such that $e^{-x}f\left(x\right)=2+{\int }_0^x\sqrt{t^4+1}dt$, for all $x\in \left(-1, 1\right)$ and let $f^{-1}$ be the inverse of $f$. Then, ${\left(f^{-1}\right)}^{\text{'}}\left(2\right)$ is equal to