What is the integral of (4x-11)/((x-5)(2x-1)) ?

The integral of (4x-11)/((x-5)(2x-1)) is ln|2x-1|+ln|2x-10|+C

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integral of (4x-11)/((x-5)(2x-1))

\int \frac{4 x - 11}{( x - 5 ) ( 2 x - 1 )} d x

ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣ + C

Solution steps

\int \frac{4 x - 11}{( x - 5 ) ( 2 x - 1 )} d x

Expand

\frac{4 x - 11}{( x - 5 ) ( 2 x - 1 )} : \frac{4 x}{( x - 5 ) ( 2 x - 1 )} - \frac{11}{( x - 5 ) ( 2 x - 1 )}

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{4 x}{( x - 5 ) ( 2 x - 1 )} d x - \int \frac{11}{( x - 5 ) ( 2 x - 1 )} d x

\int \frac{4 x}{( x - 5 ) ( 2 x - 1 )} d x = 2 (\frac{10}{9} ln ∣ 2 x - 10 ∣ - \frac{1}{9} ln ∣ 2 x - 1 ∣)

\int \frac{11}{( x - 5 ) ( 2 x - 1 )} d x = 11 (- \frac{1}{9} ln ∣ 2 x - 1 ∣ + \frac{1}{9} ln ∣ 2 x - 10 ∣)

= 2 (\frac{10}{9} ln ∣ 2 x - 10 ∣ - \frac{1}{9} ln ∣ 2 x - 1 ∣) - 11 (- \frac{1}{9} ln ∣ 2 x - 1 ∣ + \frac{1}{9} ln ∣ 2 x - 10 ∣)

Simplify

2 (\frac{10}{9} ln ∣ 2 x - 10 ∣ - \frac{1}{9} ln ∣ 2 x - 1 ∣) - 11 (- \frac{1}{9} ln ∣ 2 x - 1 ∣ + \frac{1}{9} ln ∣ 2 x - 10 ∣) : ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣

= ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣

Add a constant to the solution

= ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣ + C

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What is the integral of (4x-11)/((x-5)(2x-1)) ?
The integral of (4x-11)/((x-5)(2x-1)) is ln|2x-1|+ln|2x-10|+C