What is the integral of aln(ax) ?

The integral of aln(ax) is a(xln(ax)-x)+C

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integral of aln(ax)

\int a ln (a x) d x

a (x ln (a x) - x) + C

Solution steps

\int a ln (a x) d x

Take the constant out:

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

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

Apply Integration By Parts

: x ln (a x) - \int 1 d x

= a (x ln (a x) - \int 1 d x)

\int 1 d x = x

= a (x ln (a x) - x)

Add a constant to the solution

= a (x ln (a x) - x) + C

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