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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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