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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int \frac{u + 1}{4 u ^{\frac{1}{2}}} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{u + 1}{u ^{\frac{1}{2}}} d u

Expand

\frac{u + 1}{u ^{\frac{1}{2}}} : u^{\frac{1}{2}} + \frac{1}{u ^{\frac{1}{2}}}

= \frac{1}{4} \cdot \int u^{\frac{1}{2}} + \frac{1}{u ^{\frac{1}{2}}} 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

= \frac{1}{4} (\int u^{\frac{1}{2}} d u + \int \frac{1}{u ^{\frac{1}{2}}} d u)

\int u^{\frac{1}{2}} d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}} + 2 u^{\frac{1}{2}})

Substitute back

u = 2 x - 1

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

Simplify

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

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

Add a constant to the solution

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

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