Property 3

Q. Let $f\left(x\right)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime }\left(x\right)=f^{\prime }(2-x)$ for all $x\in (0,2),$ $f\left(0\right)=1$ and $f\left(2\right)=e^2.$ Then the value of $\underset{0}{\overset{2}{\int }}f\left(x\right)dx$ is

1. $1+e^2$
2. $1-e^2$
3. $2(1-e^2)$
4. $2(1+e^2)$

Q. If $I=\underset{0}{\overset{3\pi }{\int }}\left[\tan x\right]dx,$ where $[.]$ represents the greatest integer function, then the value of $\left|\frac{2I}{\pi }\right|$ is

Q. Find the integral $\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx$.

1. $0$
2. $1$
3. $2$
4. $\infty$

Q.

${\int }_a^b\left(\frac{\left|x\right|}{x}\right)dx,$ where $a<0<b=$

1. $\left|b\right|-\left|a\right|$

2. $\left|b\right|+\left|a\right|$

3. $\left|a-b\right|$

4. None of these

Q. The value of ${\int }_0^2\left(x^2\right]\mathrm{dx}$ is equal to

1. $5-\sqrt{2}-\sqrt{3}$
2. $5+\sqrt{2}+\sqrt{3}$
3. $5+\sqrt{2}-\sqrt{3}$
4. $5-\sqrt{2}+\sqrt{3}$

Q. If $\underset{1/8}{\overset{1/2}{\int }}\left[\ln \left[\frac{1}{x}\right]\right]dx$ is equal to $\frac{a}{b},$ where $a$ and $b$ are coprime, then the value of $b-4a$ is
($[.]$ denotes the greatest integer function)

Q. ${\int }_1^{\infty }\frac{1}{x^2}dx$ does not have a finite value

1. False
2. True