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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{3 u ^{\frac{1}{2}}}{2} d u

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

Take the constant out:

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

= \frac{3}{2} \cdot \int u^{\frac{1}{2}} d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{3}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = \frac{2}{3} x - 1

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

Simplify

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

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

Add a constant to the solution

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

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