What is the integral of ((2x-1)/(x-4)) ?

The integral of ((2x-1)/(x-4)) is 2x-8+7ln|x-4|+C

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integral of ((2x-1)/(x-4))

\int (\frac{2 x - 1}{x - 4}) d x

2 x - 8 + 7 ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{2 x - 1}{x - 4} d x

Expand

\frac{2 x - 1}{x - 4} : \frac{2 x}{x - 4} - \frac{1}{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{2 x}{x - 4} d x - \int \frac{1}{x - 4} d x

\int \frac{2 x}{x - 4} d x = 2 (x - 4 + 4 ln ∣ x - 4 ∣)

\int \frac{1}{x - 4} d x = ln ∣ x - 4 ∣

= 2 (x - 4 + 4 ln ∣ x - 4 ∣) - ln ∣ x - 4 ∣

Simplify

2 (x - 4 + 4 ln ∣ x - 4 ∣) - ln ∣ x - 4 ∣ : 2 x - 8 + 7 ln ∣ x - 4 ∣

= 2 x - 8 + 7 ln ∣ x - 4 ∣

Add a constant to the solution

= 2 x - 8 + 7 ln ∣ x - 4 ∣ + C

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What is the integral of ((2x-1)/(x-4)) ?
The integral of ((2x-1)/(x-4)) is 2x-8+7ln|x-4|+C