What is the integral of (-x)/(1-0.5x) ?

The integral of (-x)/(1-0.5x) is -4(1-0.5x-ln|1-0.5x|)+C

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integral of (-x)/(1-0.5x)

\int \frac{- x}{1 - 0.5 x} d x

- 4 (1 - 0.5 x - ln ∣ 1 - 0.5 x ∣) + C

Solution steps

\int \frac{- x}{1 - 0.5 x} d x

Take the constant out:

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

= - \int \frac{x}{1 - 0.5 x} d x

Apply u-substitution

= - \int \frac{2 ( u - 1 )}{0.5 u} d u

Take the constant out:

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

= - \frac{2}{0.5} \cdot \int \frac{u - 1}{u} d u

Simplify

= - 4 \cdot \int \frac{u - 1}{u} d u

Expand

\frac{u - 1}{u} : 1 - \frac{1}{u}

= - 4 \cdot \int 1 - \frac{1}{u} 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

= - 4 (\int 1 d u - \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= - 4 (u - ln ∣ u ∣)

Substitute back

u = 1 - 0.5 x

= - 4 (1 - 0.5 x - ln ∣ 1 - 0.5 x ∣)

Add a constant to the solution

= - 4 (1 - 0.5 x - ln ∣ 1 - 0.5 x ∣) + C

What is the integral of (-x)/(1-0.5x) ?
The integral of (-x)/(1-0.5x) is -4(1-0.5x-ln|1-0.5x|)+C