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

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

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integral of x/(3-x)

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

3 - x - 3 ln ∣ 3 - x ∣ + C

Solution steps

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

Apply u-substitution

: \int - \frac{- u + 3}{u} d u

= \int - \frac{- u + 3}{u} d u

Take the constant out:

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

= - \int \frac{- u + 3}{u} d u

Expand

\frac{- u + 3}{u} : - 1 + \frac{3}{u}

= - \int - 1 + \frac{3}{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

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

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= - (- u + 3 ln ∣ u ∣)

Substitute back

u = 3 - x

= - (- (3 - x) + 3 ln ∣ 3 - x ∣)

Simplify

= 3 - x - 3 ln ∣ 3 - x ∣

Add a constant to the solution

= 3 - x - 3 ln ∣ 3 - x ∣ + C

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