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

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

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

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

3 (x - 4 + 4 ln ∣ x - 4 ∣) + C

Solution steps

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

Apply u-substitution

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

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

Expand

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

= 3 \cdot \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 = x - 4

= 3 (x - 4 + 4 ln ∣ x - 4 ∣)

Add a constant to the solution

= 3 (x - 4 + 4 ln ∣ x - 4 ∣) + C

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