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

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

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

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

ln ∣ x ∣ + 2 x + x + C

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int 1 d x = x

= ln ∣ x ∣ + 2 x + x

Add a constant to the solution

= ln ∣ x ∣ + 2 x + x + C

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

\int (e^{y x} + y e^{y x} x + 2 y - \frac{1}{x ^{2}}) d y

\frac{d}{d x} (x (1 - x)^{6})

\int_{0}^{2} (x^{2} - 10 x) d x

\int ln (2 x) (x) d x

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