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

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

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

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

\frac{3}{5} x (x - 1)^{\frac{2}{3}} - \frac{3}{5} (x - 1)^{\frac{2}{3}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2 x}{3 ( x - 1 ) ^{\frac{1}{3}}} d x = \frac{1}{5} (2 x + 3) (x - 1)^{\frac{2}{3}}

= x (x - 1)^{\frac{2}{3}} - \frac{1}{5} (2 x + 3) (x - 1)^{\frac{2}{3}}

Simplify

x (x - 1)^{\frac{2}{3}} - \frac{1}{5} (2 x + 3) (x - 1)^{\frac{2}{3}} : \frac{3}{5} x (x - 1)^{\frac{2}{3}} - \frac{3}{5} (x - 1)^{\frac{2}{3}}

= \frac{3}{5} x (x - 1)^{\frac{2}{3}} - \frac{3}{5} (x - 1)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} x (x - 1)^{\frac{2}{3}} - \frac{3}{5} (x - 1)^{\frac{2}{3}} + C

\int x^{\frac{4}{7}} d x

\int (sin (4 x)) d x

x \to 4 lim (\frac{1}{4 - x})

\frac{d}{d x} (- sin (x^{2}) \cdot 2 x)

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

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