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Popular Calculus >

integral of x/(2-x)

Solution

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

Solution

- 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

Graph

Popular Examples

derivative of (2x^2)/((3-2x)^4)

d er i v a t i v e \frac{2 x ^{2}}{( 3 - 2 x ) ^{4}}

integral of (sec(θ))/(tan^2(θ))

\int \frac{sec ( θ )}{tan ^{2} ( θ )} d θ

d/(dθ)(cot^2(cos(θ)))

\frac{d}{d θ} (cot^{2} (cos (θ)))

derivative of (sqrt(x+1)-3)/(x-8)

d er i v a t i v e \frac{x + 1 - 3}{x - 8}

integral of 2x^{1/2}

\int 2 x^{\frac{1}{2}} d x

Frequently Asked Questions (FAQ)

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