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

The integral of (5x-2)/((2-3x)^{2/5)} is -1/9 (-25/8 (2-3x)^{8/5}+20/3 (2-3x)^{3/5})+C

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

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

- \frac{1}{9} (- \frac{25}{8} (2 - 3 x)^{\frac{8}{5}} + \frac{20}{3} (2 - 3 x)^{\frac{3}{5}}) + C

Solution steps

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

Apply u-substitution

: \int - \frac{- 5 u + 4}{9 u ^{\frac{2}{5}}} d u

= \int - \frac{- 5 u + 4}{9 u ^{\frac{2}{5}}} d u

Take the constant out:

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

= - \frac{1}{9} \cdot \int \frac{- 5 u + 4}{u ^{\frac{2}{5}}} d u

Expand

\frac{- 5 u + 4}{u ^{\frac{2}{5}}} : - 5 u^{\frac{3}{5}} + \frac{4}{u ^{\frac{2}{5}}}

= - \frac{1}{9} \cdot \int - 5 u^{\frac{3}{5}} + \frac{4}{u ^{\frac{2}{5}}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{9} (- \int 5 u^{\frac{3}{5}} d u + \int \frac{4}{u ^{\frac{2}{5}}} d u)

\int 5 u^{\frac{3}{5}} d u = \frac{25}{8} u^{\frac{8}{5}}

\int \frac{4}{u ^{\frac{2}{5}}} d u = \frac{20}{3} u^{\frac{3}{5}}

= - \frac{1}{9} (- \frac{25}{8} u^{\frac{8}{5}} + \frac{20}{3} u^{\frac{3}{5}})

Substitute back

u = 2 - 3 x

= - \frac{1}{9} (- \frac{25}{8} (2 - 3 x)^{\frac{8}{5}} + \frac{20}{3} (2 - 3 x)^{\frac{3}{5}})

Add a constant to the solution

= - \frac{1}{9} (- \frac{25}{8} (2 - 3 x)^{\frac{8}{5}} + \frac{20}{3} (2 - 3 x)^{\frac{3}{5}}) + C

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What is the integral of (5x-2)/((2-3x)^{2/5)} ?
The integral of (5x-2)/((2-3x)^{2/5)} is -1/9 (-25/8 (2-3x)^{8/5}+20/3 (2-3x)^{3/5})+C