What is the integral of x(x^{1/2}-1) ?

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

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integral of x(x^{1/2}-1)

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

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

Solution steps

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

Expand

x (x^{\frac{1}{2}} - 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{x ^{2}}{2}

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

Add a constant to the solution

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

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