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

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

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integral of (2(x+1))/x

\int \frac{2 ( x + 1 )}{x} d x

2 (x + ln ∣ x ∣) + C

Solution steps

\int \frac{2 ( x + 1 )}{x} d x

Take the constant out:

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

= 2 \cdot \int \frac{x + 1}{x} d x

Expand

\frac{x + 1}{x} : 1 + \frac{1}{x}

= 2 \cdot \int 1 + \frac{1}{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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

= 2 (x + ln ∣ x ∣)

Add a constant to the solution

= 2 (x + ln ∣ x ∣) + C