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

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

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integral of ln(3x+2)

\int ln (3 x + 2) d x

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

Solution steps

\int ln (3 x + 2) d x

Apply u-substitution

: \int ln (u) \frac{1}{3} d u

= \int ln (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int ln (u) d u

Apply Integration By Parts

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

= \frac{1}{3} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{3} (u ln (u) - u)

Substitute back

u = 3 x + 2

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

Simplify

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

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

Add a constant to the solution

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

sin^{7} (x)

\int - 2 + 12 x - 12 x^{2} d x

\int \frac{e ^{8 x}}{e ^{8 x} + 4} d x

f (x) = x^{2} + 3 x - 1, at x = 0

x \to 0 - lim (x \cdot ln (x))

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