What is the integral of (u-1)/(sqrt(u)) ?

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

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integral of (u-1)/(sqrt(u))

\int \frac{u - 1}{u} d u

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

Solution steps

\int \frac{u - 1}{u} d u

Expand

\frac{u - 1}{u} : u - \frac{1}{u}

= \int u - \frac{1}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u - \int \frac{1}{u} d u

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

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

= \frac{2}{3} u^{\frac{3}{2}} - 2 u

Add a constant to the solution

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

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