What is the integral of ((8x+2))/x ?

The integral of ((8x+2))/x is 8x+2ln|x|+C

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integral of ((8x+2))/x

integral

\frac{( 8 x + 2 )}{x}

8 x + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{8 x + 2}{x} d x

Expand

\frac{8 x + 2}{x} : 8 + \frac{2}{x}

= \int 8 + \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

= \int 8 d x + \int \frac{2}{x} d x

\int 8 d x = 8 x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= 8 x + 2 ln ∣ x ∣

Add a constant to the solution

= 8 x + 2 ln ∣ x ∣ + C