What is the integral of (3x-2)/(sqrt(4-5x)) ?

The integral of (3x-2)/(sqrt(4-5x)) is 2/25 (-5x+4)^{3/2}-4/25 sqrt(4-5x)+C

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integral of (3x-2)/(sqrt(4-5x))

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

\frac{2}{25} (- 5 x + 4)^{\frac{3}{2}} - \frac{4}{25} 4 - 5 x + C

Solution steps

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

Expand

\frac{3 x - 2}{4 - 5 x} : \frac{3 x}{4 - 5 x} - \frac{2}{4 - 5 x}

= \int \frac{3 x}{4 - 5 x} - \frac{2}{4 - 5 x} d x

Apply the Sum Rule:

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

= \int \frac{3 x}{4 - 5 x} d x - \int \frac{2}{4 - 5 x} d x

\int \frac{3 x}{4 - 5 x} d x = \frac{6}{25} (\frac{1}{3} (4 - 5 x)^{\frac{3}{2}} - 4 4 - 5 x)

\int \frac{2}{4 - 5 x} d x = - \frac{4}{5} 4 - 5 x

= \frac{6}{25} (\frac{1}{3} (4 - 5 x)^{\frac{3}{2}} - 4 4 - 5 x) - (- \frac{4}{5} 4 - 5 x)

Simplify

\frac{6}{25} (\frac{1}{3} (4 - 5 x)^{\frac{3}{2}} - 4 4 - 5 x) - (- \frac{4}{5} 4 - 5 x) : \frac{2}{25} (- 5 x + 4)^{\frac{3}{2}} - \frac{4}{25} 4 - 5 x

= \frac{2}{25} (- 5 x + 4)^{\frac{3}{2}} - \frac{4}{25} 4 - 5 x

Add a constant to the solution

= \frac{2}{25} (- 5 x + 4)^{\frac{3}{2}} - \frac{4}{25} 4 - 5 x + C

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What is the integral of (3x-2)/(sqrt(4-5x)) ?
The integral of (3x-2)/(sqrt(4-5x)) is 2/25 (-5x+4)^{3/2}-4/25 sqrt(4-5x)+C