Primitive of Sine x by Logarithm of Sine x

Theorem

$\ds \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$

Proof

We have:

\(\ds \map {\frac \d {\d x} } {\map \ln {\sin x} }\)

\(=\)

\(\ds \map {\frac \d {\map \d {\sin x} } } {\map \ln {\sin x} } \map {\frac \d {\d x} } {\sin x}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \frac {\cos x} {\sin x}\)

Derivative of Logarithm Function, Derivative of Sine Function

We also have, by Primitive of Sine Function:

$\ds \int \sin x \rd x = -\cos x + C$

So:

\(\ds \int \sin x \map \ln {\sin x} \rd x\)

\(=\)

\(\ds -\cos x \map \ln {\sin x} + \int \frac {\cos^2 x} {\sin x} \rd x\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds -\cos x \map \ln {\sin x} + \int \frac {1 - \sin^2 x} {\sin x} \rd x\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds -\cos x \map \ln {\sin x} + \int \csc x \rd x - \int \sin x \rd x\)

\(\ds \)

\(=\)

\(\ds -\cos x \map \ln {\sin x} + \ln \size {\tan \frac x 2} + \cos x + C\)

Primitive of Sine Function, Primitive of $\csc x$: Tangent Form

\(\ds \)

\(=\)

\(\ds \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C\)

$\blacksquare$