What is the integral of (ln(ax+b))/(sqrt(ax+b)) ?

The integral of (ln(ax+b))/(sqrt(ax+b)) is 1/a (2sqrt(ax+b)ln(ax+b)-4sqrt(ax+b))+C

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integral of (ln(ax+b))/(sqrt(ax+b))

\int \frac{ln ( a x + b )}{a x + b} d x

\frac{1}{a} (2 a x + b ln (a x + b) - 4 a x + b) + C

Solution steps

\int \frac{ln ( a x + b )}{a x + b} d x

Apply u-substitution

: \int \frac{ln ( u )}{a u} d u

= \int \frac{ln ( u )}{a u} d u

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int \frac{2}{u} d u = 4 u

= \frac{1}{a} (2 u ln (u) - 4 u)

Substitute back

u = a x + b

= \frac{1}{a} (2 a x + b ln (a x + b) - 4 a x + b)

Add a constant to the solution

= \frac{1}{a} (2 a x + b ln (a x + b) - 4 a x + b) + C

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What is the integral of (ln(ax+b))/(sqrt(ax+b)) ?
The integral of (ln(ax+b))/(sqrt(ax+b)) is 1/a (2sqrt(ax+b)ln(ax+b)-4sqrt(ax+b))+C