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

The integral of ln(y+x) is xln(x+y)+yln(x+y)-x-y+C

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integral of ln(y+x)

\int ln (y + x) d y

x ln (x + y) + y ln (x + y) - x - y + C

Solution steps

\int ln (y + x) d y

Apply u-substitution

: \int ln (u) d u

= \int ln (u) d u

Apply Integration By Parts

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

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

\int 1 d u = u

= u ln (u) - u

Substitute back

u = x + y

= (x + y) ln (x + y) - (x + y)

Simplify

(x + y) ln (x + y) - (x + y) : x ln (x + y) + y ln (x + y) - x - y

= x ln (x + y) + y ln (x + y) - x - y

Add a constant to the solution

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

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What is the integral of ln(y+x) ?
The integral of ln(y+x) is xln(x+y)+yln(x+y)-x-y+C