Answer
$f(x)=\displaystyle \frac{2}{3}x^{3/2}+2x^{1/2}+\frac{7}{3}$
Work Step by Step
$f'(x)=\displaystyle \frac{x+1}{x^{1/2}}=x^{1/2}+x^{-1/2}$
Using the antiderivatives table,
$f(x)=\displaystyle \frac{x^{3/2}}{1/2+1}+\frac{x^{-1/2+1}}{-1/2+1}$
$f(x)=\displaystyle \frac{2}{3}x^{3/2}+2x^{1/2}+C$
Given that $f(1)=5$, we find $C$:
$5=\displaystyle \frac{2}{3}(1)+2(1)+C$
$C=5-2-\displaystyle \frac{2}{3}=\frac{7}{3}$
$f(x)=\displaystyle \frac{2}{3}x^{3/2}+2x^{1/2}+\frac{7}{3}$
