If $I$ is the greatest of the definite integrals

${I_1} = \int_0^1 {{e^{ - x}}{{\cos }^2}x\,dx} , \,\, {I_2} = \int_0^1 {{e^{ - {x^2}}}} {\cos ^2}x\,dx$

${I_3} = \int_0^1 {{e^{ - {x^2}}}dx} ,\,\,{I_4} = \int_0^1 {{e^{ - {x^2}/2}}dx} ,$ then

Let $f:[0,1] \rightarrow[0,1]$ be a continuous function such that $x^2+(f(x))^2 \leq 1$ for all $x \in[0,1]$ and $\int_0^1 f(x) d x=\frac{\pi}{4}$ Then, $\int_{\frac{1}{2}}^{\frac{1}{\sqrt{2}}} \frac{f(x)}{1-x^2} d x$ equals

The number of continuous functions $f:[0,1] \rightarrow R$ that satisfy $\int \limits_0^1 x f(x) d x=\frac{1}{3}+\frac{1}{4} \int \limits_0^1(f(x))^2 d x$ is

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 :

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

The number of continuous functions $f :\left[0, \frac{3}{2}\right] \rightarrow(0, \infty)$ satisfying the equation $4 \int \limits_0^{3 / 2} f(x) d x+125 \int \limits_0^{3 / 2} \frac{d x}{\sqrt{f(x)+x^2}}=108$ is