What is the integral of ((sqrt(x)-1)^3)/(2sqrt(x)) ?

The integral of ((sqrt(x)-1)^3)/(2sqrt(x)) is (x^2)/4-x^{3/2}+3/2 x-sqrt(x)+C

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integral of ((sqrt(x)-1)^3)/(2sqrt(x))

\int \frac{( x - 1 ) ^{3}}{2 x} d x

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

Solution steps

\int \frac{( x - 1 ) ^{3}}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{( x - 1 ) ^{3}}{x} d x

Long division

\frac{( x - 1 ) ^{3}}{x} : x - 3 x + 3 - \frac{1}{x}

= \frac{1}{2} \cdot \int x - 3 x + 3 - \frac{1}{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

= \frac{1}{2} (\int x d x - \int 3 x d x + \int 3 d x - \int \frac{1}{x} d x)

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

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

\int 3 d x = 3 x

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

= \frac{1}{2} (\frac{x ^{2}}{2} - 2 x^{\frac{3}{2}} + 3 x - 2 x)

Simplify

\frac{1}{2} (\frac{x ^{2}}{2} - 2 x^{\frac{3}{2}} + 3 x - 2 x) : \frac{x ^{2}}{4} - x^{\frac{3}{2}} + \frac{3}{2} x - x

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

Add a constant to the solution

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

\int 4 x^{3} (x^{4} + 5)^{6} d x

x \to + 1 + lim (1 - x^{2})

\int \frac{arctan ( 7 x )}{1 + 49 x ^{2}} d x

x \to \frac{π}{3} lim (1 - 2 cos (x))

\frac{5}{x}, at x = 4

What is the integral of ((sqrt(x)-1)^3)/(2sqrt(x)) ?
The integral of ((sqrt(x)-1)^3)/(2sqrt(x)) is (x^2)/4-x^{3/2}+3/2 x-sqrt(x)+C