The value of the definite integral, $\int\limits_0^{100} {\frac{x}{{{e^{{x^2}}}}}\,dx} $ is equal to

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

If $[x]$ is the greatest integer $\leq x$, then $\pi^{2} \int_{0}^{2}\left(\sin \frac{\pi \mathrm{x}}{2}\right)(\mathrm{x}-[\mathrm{x}])^{[\mathrm{x}]} \mathrm{d} \mathrm{x}$ 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

Suppose $f(x)$ is a differentiable real function such that $f(x) + f'(x) \le 1$ for all $x$ and $f(0)=0$ . The largest possible value of $f(1)$ is

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