What is the integral of u/(sqrt(2u-1)) ?

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

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integral of u/(sqrt(2u-1))

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

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

Solution steps

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

Apply u-substitution

: \int \frac{v ^{2} + 1}{2} d v

= \int \frac{v ^{2} + 1}{2} d v

Take the constant out:

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

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

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}{2} (\int v^{2} d v + \int 1 d v)

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

\int 1 d v = v

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

Substitute back

v = 2 u - 1

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

Simplify

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

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

Add a constant to the solution

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

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