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

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

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integral of x/((2-x))

integral

\frac{x}{( 2 - x )}

- 2 ln ∣ 2 - x ∣ - x + C

Solution steps

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

\frac{x}{2 - x} = \frac{2}{2 - x} - 1

= \int \frac{2}{2 - x} - 1 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{2}{2 - x} d x - \int 1 d x

\int \frac{2}{2 - x} d x = - 2 ln ∣ 2 - x ∣

\int 1 d x = x

= - 2 ln ∣ 2 - x ∣ - x

Add a constant to the solution

= - 2 ln ∣ 2 - x ∣ - x + C

\frac{e ^{- 1}}{x}

(x - 1)^{2}

(e^{x} + e^{- x})^{4}

\frac{1}{( 1 + a ^{2} t ^{2} )}

(3^{x} + 3^{- x})^{2}