What is the integral of (3x)/(4-x) ?

The integral of (3x)/(4-x) is -3(-4+x+4ln|4-x|)+C

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integral of (3x)/(4-x)

\int \frac{3 x}{4 - x} d x

- 3 (- 4 + x + 4 ln ∣ 4 - x ∣) + C

Solution steps

\int \frac{3 x}{4 - x} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{4 - x} d x

Apply u-substitution

: \int - \frac{- u + 4}{u} d u

= 3 \cdot \int - \frac{- u + 4}{u} d u

Take the constant out:

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

= 3 (- \int \frac{- u + 4}{u} d u)

Expand

\frac{- u + 4}{u} : - 1 + \frac{4}{u}

= 3 (- \int - 1 + \frac{4}{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

= 3 (- (- \int 1 d u + \int \frac{4}{u} d u))

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= 3 (- (- u + 4 ln ∣ u ∣))

Substitute back

u = 4 - x

= 3 (- (- (4 - x) + 4 ln ∣ 4 - x ∣))

Simplify

= - 3 (- 4 + x + 4 ln ∣ 4 - x ∣)

Add a constant to the solution

= - 3 (- 4 + x + 4 ln ∣ 4 - x ∣) + C

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What is the integral of (3x)/(4-x) ?
The integral of (3x)/(4-x) is -3(-4+x+4ln|4-x|)+C