If $\int\limits_0^1 \alpha e^{\beta x^2} \sin (x+k) d x=0$ for some $\alpha, \beta \in R$, $\alpha \neq 0$, then the value of $k$ can belong to the interval

$\left[\frac{3 \pi}{4}, \frac{5 \pi}{6}\right]$

We know that the value of the integral can be zero if either the limits coincide or the integrand is identically zero or the integrand changes its sign in the interval of integration.

$\int\limits_0^1 \alpha e^{\beta x^2} \sin (x+k)=0$

$\Rightarrow \sin (x+k)$ must change its sign in [0, 1]

$\Rightarrow \sin (x+k)=0$ for same $x \in[0,1]$

$\Rightarrow k \in\left[\frac{3 \pi}{4}, \frac{5 \pi}{6}\right]$