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

The integral of-2/3 (1-x)^{3/2} is 4/15 (1-x)^{5/2}+C

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

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

\frac{4}{15} (1 - x)^{\frac{5}{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

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

Substitute back

u = 1 - x

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

Simplify

- \frac{2}{3} (- \frac{2}{5} (1 - x)^{\frac{5}{2}}) : \frac{4}{15} (1 - x)^{\frac{5}{2}}

= \frac{4}{15} (1 - x)^{\frac{5}{2}}

Add a constant to the solution

= \frac{4}{15} (1 - x)^{\frac{5}{2}} + C

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What is the integral of-2/3 (1-x)^{3/2} ?
The integral of-2/3 (1-x)^{3/2} is 4/15 (1-x)^{5/2}+C