What is the integral of (ln(sin(x))) ?

The integral of (ln(sin(x))) is i 1/2 (x^2+\Li_{2}(e^{2ix})-xln(1-(-1)^{(2x)/pi})+xln(sin(x)))+C

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integral of (ln(sin(x)))

\int (ln (sin (x))) d x

i \frac{1}{2} (x^{2} + Li_{2} (e^{2 i x}) - x ln (1 - (- 1)^{\frac{2 x}{π}}) + x ln (sin (x))) + C

Solution steps

\int ln (sin (x)) d x

Use the nonelementary integral:

\int ln (sin (x)) d x = i \frac{1}{2} (x^{2} + Li_{2} (e^{2 i x}) - x ln (1 - (- 1)^{\frac{2 x}{π}}) + x ln (sin (x)))

= i \frac{1}{2} (x^{2} + Li_{2} (e^{2 i x}) - x ln (1 - (- 1)^{\frac{2 x}{π}}) + x ln (sin (x)))

Add a constant to the solution

= i \frac{1}{2} (x^{2} + Li_{2} (e^{2 i x}) - x ln (1 - (- 1)^{\frac{2 x}{π}}) + x ln (sin (x))) + C

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What is the integral of (ln(sin(x))) ?
The integral of (ln(sin(x))) is i 1/2 (x^2+\Li_{2}(e^{2ix})-xln(1-(-1)^{(2x)/pi})+xln(sin(x)))+C