What is the integral of (4-5x)^{-3/2} ?

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

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integral of (4-5x)^{-3/2}

\int (4 - 5 x)^{- \frac{3}{2}} d x

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

Solution steps

\int (4 - 5 x)^{- \frac{3}{2}} d x

Apply u-substitution

: \int - \frac{1}{5 u ^{\frac{3}{2}}} d u

= \int - \frac{1}{5 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{1}{5} \cdot \int \frac{1}{u ^{\frac{3}{2}}} d u

Apply the Power Rule

: - \frac{2}{u ^{\frac{1}{2}}}

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

Substitute back

u = 4 - 5 x

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

Simplify

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

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

Add a constant to the solution

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

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