Answer
$\frac{(7t^2+12)^\frac{3}{2}}{21}+C$
Work Step by Step
Use the substitution $u=7t^2+12$. Then $du=14tdt$, so $t dt=\frac{1}{14}du$.
$\int t\sqrt{7t^2+12}dt$
$=\int \sqrt{u}*\frac{1}{14}du$
$=\int \frac{1}{14}\sqrt{u}du$
$=\frac{1}{14}*\frac{u^\frac{3}{2}}{\frac{3}{2}}+C$
$=\frac{u^\frac{3}{2}}{21}+C$
$=\frac{(7t^2+12)^\frac{3}{2}}{21}+C$