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

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

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

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

2 x - ln ∣ x + 1 ∣ + C

Solution steps

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

Long division

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

f (x) = \frac{3}{x ^{2}}

\frac{\partial}{\partial x} (\frac{( a + c x )}{c + b})

\int cos (θ) sin (θ) d θ

\frac{d y}{d x} = 5 x + y - 5

\frac{\partial}{\partial u} (u sin (v))

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