What is the integral of (2x-4+4sqrt(x))/(sqrt(x)) ?

The integral of (2x-4+4sqrt(x))/(sqrt(x)) is 4/3 x^{3/2}-8sqrt(x)+4x+C

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integral of (2x-4+4sqrt(x))/(sqrt(x))

\int \frac{2 x - 4 + 4 x}{x} d x

\frac{4}{3} x^{\frac{3}{2}} - 8 x + 4 x + C

Solution steps

\int \frac{2 x - 4 + 4 x}{x} d x

Expand

\frac{2 x - 4 + 4 x}{x} : 2 x - \frac{4}{x} + 4

= \int 2 x - \frac{4}{x} + 4 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 2 x d x - \int \frac{4}{x} d x + \int 4 d x

\int 2 x d x = \frac{4}{3} x^{\frac{3}{2}}

\int \frac{4}{x} d x = 8 x

\int 4 d x = 4 x

= \frac{4}{3} x^{\frac{3}{2}} - 8 x + 4 x

Add a constant to the solution

= \frac{4}{3} x^{\frac{3}{2}} - 8 x + 4 x + C

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What is the integral of (2x-4+4sqrt(x))/(sqrt(x)) ?
The integral of (2x-4+4sqrt(x))/(sqrt(x)) is 4/3 x^{3/2}-8sqrt(x)+4x+C