What is the integral of ln(a+x) ?

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

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integral of ln(a+x)

\int ln (a + x) d x

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

Solution steps

\int ln (a + x) d x

Apply u-substitution

= \int ln (u) d u

Apply Integration By Parts

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

\int 1 d u = u

= u ln (u) - u

Substitute back

u = a + x

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

Simplify

(a + x) ln (a + x) - (a + x) : a ln (a + x) + x ln (a + x) - a - x

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

Add a constant to the solution

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

What is the integral of ln(a+x) ?
The integral of ln(a+x) is aln(a+x)+xln(a+x)-a-x+C