$\int \frac32 x^\frac12 \: dx = \frac32 \cdot \fracx^\frac12+1\frac12+1 = \frac32 \cdot \fracx^\frac32\frac32 + C = x^\frac32+C$ $\int \sqrtx \: dx = \int x^\frac12 \: dx= \fracx^\frac12+1\frac12+1 = \fracx^\frac32\frac32 + C$ $\int \sqrt3x \: dx = \int (3x)^\frac12 \: dx = \int 3^\frac12 \cdot (…)