Let $I_1 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}\sin (x)dx} $ ; $I_2 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}dx} $ ; $I_3 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}(1 + x)\,dx} $

and consider the statements

$I\,:$ $I_1 < I_2$   

$II\,:$  $I_2 < I_3$ 

$III\,:$  $I_1 = I_3$

Which of the following is $(are)$ true?

The points of intersection of
${F_1}(x) = \int_2^x {(2t - 5)\,dt} $ and ${F_2}(x) = \int_0^x {2t\,dt,} $ are

If for all real triplets $(a, b, c), f(x)=a+b x+c x^{2}$ then $\int \limits_{0}^{1} f(\mathrm{x}) \mathrm{d} \mathrm{x}$ is equal to 

Let $a, b, c$ be non-zero real numbers such that ; $\int\limits_0^1 {} (1 + cos^8x) (ax^2 + bx + c) dx$ $= \int\limits_0^2 {} (1 + cos^8x) (ax^2 + bx + c) dx$ , then the quadratic equation $ax^2 + bx + c = 0$ has :

Number of values of $x$ satisfying the equation

$\int\limits_{ - \,1}^x {\,\left( {8{t^2} + \frac{{28}}{3}t + 4} \right)\,dt} $ $=$ $\frac{{\left( {{\textstyle{3 \over 2}}} \right)x + 1}}{{{{\log }_{(x + 1)}}\sqrt {x + 1} }}$ , is

Let $J=\int_0^1 \frac{x}{1+x^8} d x$

Consider the following assertions:

$I$. $J>\frac{1}{4}$

$II$. $J<\frac{\pi}{8}$ Then,