What is the integral of (1+sqrt(3x))/(sqrt(x)) ?

The integral of (1+sqrt(3x))/(sqrt(x)) is 2sqrt(x)+sqrt(3)x+C

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integral of (1+sqrt(3x))/(sqrt(x))

\int \frac{1 + 3 x}{x} d x

2 x + 3 x + C

Solution steps

\int \frac{1 + 3 x}{x} d x

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \int \frac{1 + 3 x}{x} d x

Expand

\frac{1 + 3 x}{x} : \frac{1}{x} + 3

= \int \frac{1}{x} + 3 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 3 d x

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

\int 3 d x = 3 x

= 2 x + 3 x

Add a constant to the solution

= 2 x + 3 x + C

What is the integral of (1+sqrt(3x))/(sqrt(x)) ?
The integral of (1+sqrt(3x))/(sqrt(x)) is 2sqrt(x)+sqrt(3)x+C