The smallest interval $[a,\,\,b]$ such that $\int_0^1 {\frac{{dx}}{{\sqrt {1 + {x^4}} }}} \in [a,\,\,b]$ is given by

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

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

If $f(x)$ is a quadratic in $x$ , then $\int\limits_0^1 {f(x) dx}$ 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 :

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function such that $\int_0^1 f(x) d x=10$. Which of the following statements is NOT necessarily true?