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

The integral of sqrt(x)(x-1) is 2/5 x^{5/2}-2/3 x^{3/2}+C

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integral of sqrt(x)(x-1)

\int x (x - 1) d x

\frac{2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

\int x (x - 1) d x

Expand

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

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

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

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

= \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C

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

x \to \frac{π}{3} lim (cot (3 x))

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

\frac{d}{d x} (- ln (1 + 6 x))

\frac{\partial}{\partial x} (cos^{n} (x))

What is the integral of sqrt(x)(x-1) ?
The integral of sqrt(x)(x-1) is 2/5 x^{5/2}-2/3 x^{3/2}+C