What is the integral of ln(2x+3) ?

The integral of ln(2x+3) is xln(2x+3)+3/2 ln(2x+3)-x-3/2+C

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integral of ln(2x+3)

\int ln (2 x + 3) d x

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

Solution steps

\int ln (2 x + 3) d x

Apply u-substitution

= \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

= \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 + 3

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of ln(2x+3) ?
The integral of ln(2x+3) is xln(2x+3)+3/2 ln(2x+3)-x-3/2+C