Primitive of Logarithm of x/Corollary

Corollary to Primitive of $\map \ln x$

$\ds \int \map \ln {1 - x} \rd x = \paren {x - 1} \map \ln {1 - x} - x + C$

Proof

Let $z = \paren {1 - x}$.

\(\ds z\)

\(=\)

\(\ds 1 - x\)

\(\ds \leadsto \ \ \)

\(\ds \d z\)

\(=\)

\(\ds - \rd x\)

\(\ds \leadsto \ \ \)

\(\ds \int \map \ln {1 - x} \rd x\)

\(=\)

\(\ds -\int \map \ln z \rd z\)

Integration by Substitution: $z = 1 - x$

\(\ds \)

\(=\)

\(\ds - \paren {z \ln z - z} + C\)

Primitive of Logarithm of x

\(\ds \)

\(=\)

\(\ds - z \ln z + z + C\)

\(\ds \)

\(=\)

\(\ds -\paren {1 - x} \map \ln {1 - x} + \paren {1 - x} + C\)

\(\ds \)

\(=\)

\(\ds \paren {x - 1} \map \ln {1 - x} - x + C\)

$\blacksquare$