Let $f (x) = \frac{{\sin x}}{x}$ , then $\int\limits_0^{\frac{\pi }{2}} {f(x)\,\,f\left( {\frac{\pi }{2} - x} \right)\,dx} =$

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

The minimum value of the function $f(x)=\int \limits_0^2 e^{|x-t|} d t$ is

Let for $x \in R , S_0( x )= x$,$S _{ k }( x )= C _{ k } x + k \int _0^{ x } S _{ k -1}(t) d t$, where $C _0=1, C _{ k }=1-\int_0^1 S _{ k -1}( x ) dx , k =1,2,3 \ldots$. Then $S _2(3)+6 C _3$ is equal to $...........$.

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 $f: R \rightarrow R$ be a function defined as $f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in R$, where [t] is the greatest integer less than or equal to $t$. If $\lim _{x \rightarrow-1} f(x)$ exists, then the value of $\int_{0}^{4} f(x) d x$ is equal to.