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

The integral of ((sqrt(x)-2)^2)/x is x-8sqrt(x)+4ln|x|+C

integral of ((sqrt(x)-2)^2)/x

\int \frac{( x - 2 ) ^{2}}{x} d x

x - 8 x + 4 ln ∣ x ∣ + C

Solution steps

\int \frac{( x - 2 ) ^{2}}{x} d x

Expand

\frac{( x - 2 ) ^{2}}{x} : 1 - \frac{4}{x} + \frac{4}{x}

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

\int 1 d x = x

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

= x - 8 x + 4 ln ∣ x ∣

Add a constant to the solution

= x - 8 x + 4 ln ∣ x ∣ + C

\frac{\partial}{\partial x} (5^{2 x})

\int \frac{1}{x 1 - ( ln ( x ) ) ^{2}} d x

in v erse l a pl a ce \frac{s}{( s + 4 ) ^{2}}

a re a y = x, y = 0, y = x - 2

x \to 3 + lim (\frac{2}{x - 3})

What is the integral of ((sqrt(x)-2)^2)/x ?
The integral of ((sqrt(x)-2)^2)/x is x-8sqrt(x)+4ln|x|+C