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

The integral of x^{2/3}-3x^{-3}+x^{1/2} is 3/5 x^{5/3}+3/(2x^2)+2/3 x^{3/2}+C

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integral of x^{2/3}-3x^{-3}+x^{1/2}

\int x^{\frac{2}{3}} - 3 x^{- 3} + x^{\frac{1}{2}} d x

\frac{3}{5} x^{\frac{5}{3}} + \frac{3}{2 x ^{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

\int x^{\frac{2}{3}} - 3 x^{- 3} + 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{2}{3}} d x - \int 3 x^{- 3} d x + \int x^{\frac{1}{2}} d x

\int x^{\frac{2}{3}} d x = \frac{3}{5} x^{\frac{5}{3}}

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

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

= \frac{3}{5} x^{\frac{5}{3}} - (- \frac{3}{2 x ^{2}}) + \frac{2}{3} x^{\frac{3}{2}}

Simplify

= \frac{3}{5} x^{\frac{5}{3}} + \frac{3}{2 x ^{2}} + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{3}{5} x^{\frac{5}{3}} + \frac{3}{2 x ^{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

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What is the integral of x^{2/3}-3x^{-3}+x^{1/2} ?
The integral of x^{2/3}-3x^{-3}+x^{1/2} is 3/5 x^{5/3}+3/(2x^2)+2/3 x^{3/2}+C