Question

Integrate x+1/sqrt(x). I know that the answer is 2/3 • sqrt(x) • (x+3) + C, but I don't know how it is simplified from 2/3 • x^3/2 + 2 • x^1/2 + C. Please explain this step in detail? Thank you!

Answer 1

$\displaystyle\int\frac{x+1}{\sqrt x}\,\mathrm dx=\int\frac x{x^{1/2}}+\frac1{x^{1/2}}\,\mathrm dx$

$=\displaystyle\int x^{1/2}+x^{-1/2}\,\mathrm dx$

By the power rule,

$=\dfrac{x^{3/2}}{\frac32}+\dfrac{x^{1/2}}{\frac12}+C$

(this seems to be the step you're not getting?)

$=\dfrac23x^{3/2}+2x^{1/2}+C$

The next step is to pull out a common factor of $x^{1/2}$ from the antiderivative:

$x^{3/2}=x^{1/2+1}=x^{1/2}\cdot x^1$

so that the final result is

$\dfrac23x^{3/2}+2x^{1/2}+C=\dfrac23x^{1/2}(x+3)+C$