[the integral of] (5t*[the square root of](t^2+3)i + sin(pi*t)j + 5t*[the square root of](t)k)dt
Do component wise integration:
For $\displaystyle \int 5t\sqrt{t^2+3}\,dt$ use the substitution $\displaystyle u=t^2+3$,
$\displaystyle \int\sin\left(\pi t\right)\,dt=-\frac{1}{\pi}\cos\left(\pi t\right)+C_y$
$\displaystyle \int 5t\sqrt{t}\,dt=\int 5t^{3/2}\,dt=\dots$
Can you take it from here?