Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.6 Logarithmic And Other Functions Defined By Integrals - Exercises Set 6.6 - Page 460: 36

$$\dfrac{2x^{3/2}}{2}+2 \sqrt x-\dfrac{8}{3} $$

$y(x)=y_0+\int_{x_0}^x f(t) \ dt $ We have: $y(x)=0+\int_{1}^{x} \dfrac{t+1}{\sqrt t} \ dt \\=\dfrac{t^{3/2}}{3/2} +2 t^{1/2}]_{1}^{x}\\=\dfrac{2}{3} (x^{3/2}-1)+2 (\sqrt x-1) \\=\dfrac{2x^{3/2}}{2}+2 \sqrt x-\dfrac{8}{3}$