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

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

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

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

- x - ln ∣ x - 4 ∣ + 4 + C

Solution steps

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

Expand

\frac{3 - x}{x - 4} : \frac{3}{x - 4} - \frac{x}{x - 4}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{3}{x - 4} d x = 3 ln ∣ x - 4 ∣

\int \frac{x}{x - 4} d x = x - 4 + 4 ln ∣ x - 4 ∣

= 3 ln ∣ x - 4 ∣ - (x - 4 + 4 ln ∣ x - 4 ∣)

Simplify

= - x - ln ∣ x - 4 ∣ + 4

Add a constant to the solution

= - x - ln ∣ x - 4 ∣ + 4 + C

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