Amit M Agarwal Solutions for Chapter: Definite Integrals, Exercise 2: Work Book Exercise 10.2

The value of ${\int }_0^{\frac{32\pi }{3}}\sqrt{1+\cos 2x}dx$ is

The value of ${\int }_0^{{\sin }^{2}x}{\sin }^{-1}\sqrt{t}dt+{\int }_0^{{\cos }^{2}x}{\cos }^{-1}\sqrt{t}dt$ is

The value of ${\int }_0^1{e}^{x^2}dx$ lies in the interval

If $\varphi \left(x\right)={\int }_{\frac{1}{x}}^{\sqrt{x}}\sin \left(t^2\right)dt,$ then ${\varphi }^{\text{'}}\left(1\right)$ is equal to

Let $f\left(a\right)>0$ and $f\left(x\right)$ be a non-decreasing continuous function in $[a,b].$ Then, $\frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx$ has

The smallest interval $[a,b]$ such that ${\int }_0^1\frac{1}{\sqrt{1+x^4}}dx\in \left[a,b\right],$ is given by

If $f\left(x\right)={\int }_0^{x}f\left(t\right)dt,$ then the value of $f\left(\ln 5\right)$ is

The least value of the function $\varphi \left(x\right)={\int }_{\frac{5\pi }{4}}^x\left(3\sin t+4\cos t\right)dt$ on the interval $\left[\frac{5\pi }{4},\frac{4\pi }{3}\right]$ is