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.

A quadratic polynomial $P(x)$ satisfies the conditions, $P(0) = P(1) = 0\, \&\,\int\limits_0^1 {}  P(x) dx = 1$. The leading coefficient of the quadratic polynomial is :

The number of continuous functions $f:[0,1] \rightarrow(-\infty, \infty)$ satisfying the condition $\int \limits_0^1(f(x))^2 dx =2 \int_0^1 f( x ) dx$ is

Let $a, b$ and $c$ be positive constants. The value of $‘a’$ in terms of $‘c’$ if the value of integral $\int\limits_0^1 {(ac{x^{b + 1}} + {a^3}b{x^{3b + 5}})\,dx} $ is independent of $b$ equals

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

$I=\int \limits_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x$. Then