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

The integral of x/(2-3x) is -1/3 x-2/9 ln|-3x+2|+C

Popular >

integral of x/(2-3x)

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

- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣ + C

Solution steps

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

Long division

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

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

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{2}{3 ( - 3 x + 2 )} d x = - \frac{2}{9} ln ∣ - 3 x + 2 ∣

= - \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣

Add a constant to the solution

= - \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣ + C

\frac{d}{d x} (\frac{1}{( 3 x + 8 ) ^{2}})

\frac{\partial}{\partial x} (\frac{3 x}{y})

x \to - 1 - lim (x + 2)

\int \frac{1 - 3 x}{3 + 2 x} d x

x \to 0 lim (\frac{x ^{3} - x}{x})

What is the integral of x/(2-3x) ?
The integral of x/(2-3x) is -1/3 x-2/9 ln|-3x+2|+C