What is the integral of (6x)/(3-x) ?

The integral of (6x)/(3-x) is -6(-3+x+3ln|3-x|)+C

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integral of (6x)/(3-x)

\int \frac{6 x}{3 - x} d x

- 6 (- 3 + x + 3 ln ∣ 3 - x ∣) + C

Solution steps

\int \frac{6 x}{3 - x} d x

Take the constant out:

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

= 6 \cdot \int \frac{x}{3 - x} d x

Apply u-substitution

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

= 6 \cdot \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

= 6 (- \int \frac{- u + 3}{u} d u)

Expand

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

= 6 (- \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

= 6 (- (- \int 1 d u + \int \frac{3}{u} d u))

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= 6 (- (- u + 3 ln ∣ u ∣))

Substitute back

u = 3 - x

= 6 (- (- (3 - x) + 3 ln ∣ 3 - x ∣))

Simplify

= - 6 (- 3 + x + 3 ln ∣ 3 - x ∣)

Add a constant to the solution

= - 6 (- 3 + x + 3 ln ∣ 3 - x ∣) + C

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What is the integral of (6x)/(3-x) ?
The integral of (6x)/(3-x) is -6(-3+x+3ln|3-x|)+C