What is the integral of ln(5x+8) ?

The integral of ln(5x+8) is xln(5x+8)+8/5 ln(5x+8)-x-8/5+C

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integral of ln(5x+8)

\int ln (5 x + 8) d x

x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5} + C

Solution steps

\int ln (5 x + 8) d x

Apply u-substitution

= \int ln (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int ln (u) d u

Apply Integration By Parts

= \frac{1}{5} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{5} (u ln (u) - u)

Substitute back

u = 5 x + 8

= \frac{1}{5} ((5 x + 8) ln (5 x + 8) - (5 x + 8))

Simplify

\frac{1}{5} ((5 x + 8) ln (5 x + 8) - (5 x + 8)) : x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5}

= x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5}

Add a constant to the solution

= x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5} + C

What is the integral of ln(5x+8) ?
The integral of ln(5x+8) is xln(5x+8)+8/5 ln(5x+8)-x-8/5+C