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

The integral of (5x-2)/(x(-2)) is -5/2 x+ln|x|+C

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integral of (5x-2)/(x(-2))

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

- \frac{5}{2} x + ln ∣ x ∣ + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{- 2} \cdot \int \frac{5 x - 2}{x} d x

Simplify

= (- \frac{1}{2}) \cdot \int \frac{5 x - 2}{x} d x

Expand

\frac{5 x - 2}{x} : 5 - \frac{2}{x}

= (- \frac{1}{2}) \cdot \int 5 - \frac{2}{x} d x

Apply the Sum Rule:

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

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

\int 5 d x = 5 x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= (- \frac{1}{2}) (5 x - 2 ln ∣ x ∣)

Simplify

(- \frac{1}{2}) (5 x - 2 ln ∣ x ∣) : - \frac{5}{2} x + ln ∣ x ∣

= - \frac{5}{2} x + ln ∣ x ∣

Add a constant to the solution

= - \frac{5}{2} x + ln ∣ x ∣ + C

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