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 :

If $\frac{d}{{dx}}\,G\left( x \right) = \frac{{{e^{\tan \,x}}}}{x},\,x \in \left( {0,\pi /2} \right)$, then $\int\limits_{1/4}^{1/2} {\frac{2}{x}} .{e^{\tan \,\left( {\pi \,{x^2}} \right)}}dx$ is equal to

Let the function $f :[0,2] \rightarrow R$ be defined as

$f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.$

where [t] denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int \limits_0^2 x f(x) d x$ is

The value of integral $\int_0^1 {{e^{{x^2}}}} dx$ lies in interval

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

Let $\frac{d}{{dx}}F(x) = \left( {\frac{{{e^{\sin x}}}}{x}} \right)\,;\,x > 0$. If $\int_{\,1}^{\,4} {\frac{3}{x}{e^{\sin {x^3}}}dx = F(k) - F(1)} $, then one of the possible value of $k$, is