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

The integral of (x^{1/3}-x^{2/3}) is 3/4 x^{4/3}-3/5 x^{5/3}+C

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integral of (x^{1/3}-x^{2/3})

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

\frac{3}{4} x^{\frac{4}{3}} - \frac{3}{5} x^{\frac{5}{3}} + C

Solution steps

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

\int x^{\frac{1}{3}} d x = \frac{3}{4} x^{\frac{4}{3}}

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

= \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{5} x^{\frac{5}{3}}

Add a constant to the solution

= \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{5} x^{\frac{5}{3}} + C

What is the integral of (x^{1/3}-x^{2/3}) ?
The integral of (x^{1/3}-x^{2/3}) is 3/4 x^{4/3}-3/5 x^{5/3}+C