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

The integral of (a^{2/3}-x^{2/3})^3 is a^2x-9/5 a^{4/3}x^{5/3}+9/7 a^{2/3}x^{7/3}-(x^3)/3+C

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

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

a^{2} x - \frac{9}{5} a^{\frac{4}{3}} x^{\frac{5}{3}} + \frac{9}{7} a^{\frac{2}{3}} x^{\frac{7}{3}} - \frac{x ^{3}}{3} + C

Solution steps

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

Expand

(a^{\frac{2}{3}} - x^{\frac{2}{3}})^{3} : a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} - x^{2}

= \int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} - x^{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 a^{2} d x - \int 3 a^{\frac{4}{3}} x^{\frac{2}{3}} d x + \int 3 a^{\frac{2}{3}} x^{\frac{4}{3}} d x - \int x^{2} d x

\int a^{2} d x = a^{2} x

\int 3 a^{\frac{4}{3}} x^{\frac{2}{3}} d x = \frac{9}{5} a^{\frac{4}{3}} x^{\frac{5}{3}}

\int 3 a^{\frac{2}{3}} x^{\frac{4}{3}} d x = \frac{9}{7} a^{\frac{2}{3}} x^{\frac{7}{3}}

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

= a^{2} x - \frac{9}{5} a^{\frac{4}{3}} x^{\frac{5}{3}} + \frac{9}{7} a^{\frac{2}{3}} x^{\frac{7}{3}} - \frac{x ^{3}}{3}

Add a constant to the solution

= a^{2} x - \frac{9}{5} a^{\frac{4}{3}} x^{\frac{5}{3}} + \frac{9}{7} a^{\frac{2}{3}} x^{\frac{7}{3}} - \frac{x ^{3}}{3} + C

\frac{1}{x - 1}, 9

\int \frac{tan ( x )}{cos ^{2} ( x )} d x

x \to \frac{π}{2} lim (7 sin (x))

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

\frac{\partial}{\partial x} (\frac{x ^{2} y ^{2}}{x ^{2} + y ^{2}})

What is the integral of (a^{2/3}-x^{2/3})^3 ?
The integral of (a^{2/3}-x^{2/3})^3 is a^2x-9/5 a^{4/3}x^{5/3}+9/7 a^{2/3}x^{7/3}-(x^3)/3+C