Answer
$G(t) = 2t^{\frac{1}{2}} + \frac{2}{3}t^{\frac{3}{2}} + \frac{2}{5}t^{\frac{5}{2}} + C$
Work Step by Step
$g(t) = \frac{1 + t + t^{2}}{\sqrt t}$
Expand the function:
$g(t) = \frac{1}{\sqrt t} + \frac{t}{\sqrt t} + \frac{t^{2}}{\sqrt t}$
Convert the square roots into exponents:
$g(t) = \frac{1}{t^{\frac{1}{2}}} + \frac{t}{t^{\frac{1}{2}}} + \frac{t^{2}}{t^{\frac{1}{2}}}$
Simplify:
$g(t) = t^{\frac{-1}{2}} + t^{\frac{1}{2}} + t^{\frac{3}{2}}$
Apply the Power rule: $\frac{x^{a+1}}{a+1}$:
$G(t) = \frac{t^{\frac{-1}{2}+1}}{-\frac{1}{2}+1} + \frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1} + \frac{t^{\frac{3}{2}+1}}{\frac{3}{2}+1}$
$G(t) = \frac{t^{\frac{1}{2}}}{\frac{1}{2}} + \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + \frac{t^{\frac{5}{2}}}{\frac{5}{2}}$
$G(t) = 2t^{\frac{1}{2}} + \frac{2}{3}t^{\frac{3}{2}} + \frac{2}{5}t^{\frac{5}{2}}$
Add the constant to the solution:
$G(t) = 2t^{\frac{1}{2}} + \frac{2}{3}t^{\frac{3}{2}} + \frac{2}{5}t^{\frac{5}{2}} + C$
