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

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

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

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

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

Solution steps

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

Long division

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

= \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

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

Add a constant to the solution

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

\frac{d}{d x} (A (e^{2 x} + 2 e^{2 x} x))

\frac{d}{d x} (- 2 cot (x) + 6 x^{4} csc (x))

f (x) = x^{\frac{1}{9}}

\frac{d}{d x} (8 - 5 x)

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

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