What is the integral of (5+sqrt(x)+x)/x ?

The integral of (5+sqrt(x)+x)/x is 5ln|x|+2sqrt(x)+x+C

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integral of (5+sqrt(x)+x)/x

\int \frac{5 + x + x}{x} d x

5 ln ∣ x ∣ + 2 x + x + C

Solution steps

\int \frac{5 + x + x}{x} d x

Expand

\frac{5 + x + x}{x} : \frac{5}{x} + \frac{1}{x} + 1

= \int \frac{5}{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{5}{x} d x + \int \frac{1}{x} d x + \int 1 d x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

\int 1 d x = x

= 5 ln ∣ x ∣ + 2 x + x

Add a constant to the solution

= 5 ln ∣ x ∣ + 2 x + x + C

\int \frac{( 3 + x + x ^{2} )}{x} d x

x \to 2 lim (x^{2} - 2 x + 4)

\int cos (y x) d x

x \frac{d y}{d x} + 2 y = x^{- 3}

y = e^{2 x} + e^{- 2 x}

What is the integral of (5+sqrt(x)+x)/x ?
The integral of (5+sqrt(x)+x)/x is 5ln|x|+2sqrt(x)+x+C