What is the integral of ((4+sqrt(x)-3x))/x ?

The integral of ((4+sqrt(x)-3x))/x is 4ln|x|+2sqrt(x)-3x+C

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integral of ((4+sqrt(x)-3x))/x

\int \frac{( 4 + x - 3 x )}{x} d x

4 ln ∣ x ∣ + 2 x - 3 x + C

Solution steps

\int \frac{4 + x - 3 x}{x} d x

Expand

\frac{4 + x - 3 x}{x} : \frac{4}{x} + \frac{1}{x} - 3

= \int \frac{4}{x} + \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{4}{x} d x + \int \frac{1}{x} d x - \int 3 d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

\int 3 d x = 3 x

= 4 ln ∣ x ∣ + 2 x - 3 x

Add a constant to the solution

= 4 ln ∣ x ∣ + 2 x - 3 x + C

What is the integral of ((4+sqrt(x)-3x))/x ?
The integral of ((4+sqrt(x)-3x))/x is 4ln|x|+2sqrt(x)-3x+C