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

The true set of values of $‘a’$ for which the inequality $\int\limits_a^0 {} (3^{ -2x} - 2. 3^{-x})\, dx \geq 0$ is true 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

If for all real triplets $(a, b, c), f(x)=a+b x+c x^{2}$ then $\int \limits_{0}^{1} f(\mathrm{x}) \mathrm{d} \mathrm{x}$ is equal to 

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, \mathrm{f}(1))$ and $(3, \mathrm{f}(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3} \quad$ where $\alpha, \quad \beta$ are integers, then the value of $\alpha+\beta$ 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