Expectation of Brownian motion and indicator function

Question

I want to calculate the expression $$ \mathbb{E}[B_t{1}_{\{B_t<a\}}] $$ where $B_t$ is a standard Brownian motion, my instinct tells me that this is just $$ \int_{-\infty}^a x\dfrac{1}{\sqrt{2\pi t}}e^{-\frac{1}{2}\frac{x^2}{t}}\,dx=-e^{-\frac{a^2}{2t}}\sqrt{\frac{t}{2\pi}}. $$ Is this right?