What is the integral of (x^{1/2}+x^{-1/2}) ?

The integral of (x^{1/2}+x^{-1/2}) is 2/3 x^{3/2}+2x^{1/2}+C

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integral of (x^{1/2}+x^{-1/2})

\int (x^{\frac{1}{2}} + x^{- \frac{1}{2}}) d x

\frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C

Solution steps

\int x^{\frac{1}{2}} + x^{- \frac{1}{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{1}{2}} d x + \int x^{- \frac{1}{2}} d x

\int x^{\frac{1}{2}} d x = \frac{2}{3} x^{\frac{3}{2}}

\int x^{- \frac{1}{2}} d x = 2 x^{\frac{1}{2}}

= \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C

What is the integral of (x^{1/2}+x^{-1/2}) ?
The integral of (x^{1/2}+x^{-1/2}) is 2/3 x^{3/2}+2x^{1/2}+C