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

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

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

integral

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

2 x - ln ∣ x + 1 ∣ + C

Solution steps

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

Long division

\frac{1 + 2 x}{1 + x} : 2 - \frac{1}{x + 1}

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

\int 2 d x = 2 x

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

= 2 x - ln ∣ x + 1 ∣

Add a constant to the solution

= 2 x - ln ∣ x + 1 ∣ + C

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