What is the integral of ln(2x-1) ?

The integral of ln(2x-1) is xln(2x-1)-1/2 ln(2x-1)-x+1/2+C

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integral of ln(2x-1)

\int ln (2 x - 1) d x

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

Solution steps

\int ln (2 x - 1) d x

Apply u-substitution

: \int ln (u) \frac{1}{2} d u

= \int ln (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}{2} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{2} (u ln (u) - u)

Substitute back

u = 2 x - 1

= \frac{1}{2} ((2 x - 1) ln (2 x - 1) - (2 x - 1))

Simplify

\frac{1}{2} ((2 x - 1) ln (2 x - 1) - (2 x - 1)) : x ln (2 x - 1) - \frac{1}{2} ln (2 x - 1) - x + \frac{1}{2}

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

Add a constant to the solution

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

\frac{d}{d x} (2 x^{3} - 3 x)

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\int 4 x^{3} - 3 x^{2} + 5 d x

What is the integral of ln(2x-1) ?
The integral of ln(2x-1) is xln(2x-1)-1/2 ln(2x-1)-x+1/2+C