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

f (x) = - 2 x^{4} + 5 x^{2} - 3, (1, 0)

\frac{d ^{2}}{d x ^{2}} (\frac{1}{x})

\frac{\partial}{\partial x} (\frac{y}{2 x + y})

\frac{\partial}{\partial x} (6 x y e^{x z})

x \to 0 lim (\frac{tan ( \frac{x}{3} )}{x})

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