If `int_(0)^(1) alphae^(betax^(2))sin(x+k)dx=0` for some `alpha,beta in R,alpha ne 0`, then the value of k can belong to the interval

Options

A. `[(pi)/(3),(5pi)/(12)]`
B. `[(pi)/(3),(pi)/(2)]`
C. `[(3pi)/(4),(5pi)/(6)]`
D. `[-(pi)/(2),-(pi)/(3)]`

Answer

C

Explanation

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.
`underset(0)overset(1)int alpha e^(betax^(2))sin(x+k)=0`
`rArr sin (x+k)` must changes its sign in [0,1].
`rArr sin(x+k)=0` for same `x in [0,1]`
`rArr k in [(3pi)/(4),(5pi)/(6)]`