What is the integral of (x-7)/(x(2x-7)) ?

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

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integral of (x-7)/(x(2x-7))

\int \frac{x - 7}{x ( 2 x - 7 )} d x

- \frac{1}{2} ln ∣ 2 x - 7 ∣ + ln ∣ x ∣ + C

Solution steps

\int \frac{x - 7}{x ( 2 x - 7 )} d x

Expand

\frac{x - 7}{x ( 2 x - 7 )} : \frac{x}{x ( 2 x - 7 )} - \frac{7}{x ( 2 x - 7 )}

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{x}{x ( 2 x - 7 )} d x - \int \frac{7}{x ( 2 x - 7 )} d x

\int \frac{x}{x ( 2 x - 7 )} d x = \frac{1}{2} ln ∣ 2 x - 7 ∣

\int \frac{7}{x ( 2 x - 7 )} d x = - ln ∣ x ∣ + ln ∣ 2 x - 7 ∣

= \frac{1}{2} ln ∣ 2 x - 7 ∣ - (- ln ∣ x ∣ + ln ∣ 2 x - 7 ∣)

Simplify

\frac{1}{2} ln ∣ 2 x - 7 ∣ - (- ln ∣ x ∣ + ln ∣ 2 x - 7 ∣) : - \frac{1}{2} ln ∣ 2 x - 7 ∣ + ln ∣ x ∣

= - \frac{1}{2} ln ∣ 2 x - 7 ∣ + ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ 2 x - 7 ∣ + ln ∣ x ∣ + C

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