integral of (ln(x))/(sqrt(1-x^2))

Lösung

\int \frac{ln ( x )}{1 - x ^{2}} d x

i \frac{1}{2} (arcsin^{2} (x) + Li_{2} (e^{2 i a r c s i n (x)}) - arcsin (x) ln (1 - (- 1)^{\frac{2}{π} a r c s i n (x)}) + arcsin (x) ln (x)) + C

Schritte zur Lösung

\int \frac{ln ( x )}{1 - x ^{2}} d x

Trigonometrische Substitution anwenden

= \int ln (sin (u)) d u

Das ist ein nicht elementares 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} (u^{2} + Li_{2} (e^{2 i u}) - u ln (1 - (- 1)^{\frac{2 u}{π}}) + u ln (sin (u)))

Setze in

u = arcsin (x)

ein

= i \frac{1}{2} (arcsin^{2} (x) + Li_{2} (e^{2 i a r c s i n (x)}) - arcsin (x) ln (1 - (- 1)^{\frac{2 a r c s i n ( x )}{π}}) + arcsin (x) ln (sin (arcsin (x))))

Vereinfache

i \frac{1}{2} (arcsin^{2} (x) + Li_{2} (e^{2 i a r c s i n (x)}) - arcsin (x) ln (1 - (- 1)^{\frac{2 a r c s i n ( x )}{π}}) + arcsin (x) ln (sin (arcsin (x)))) : i \frac{1}{2} (arcsin^{2} (x) + Li_{2} (e^{2 i a r c s i n (x)}) - arcsin (x) ln (1 - (- 1)^{\frac{2}{π} a r c s i n (x)}) + arcsin (x) ln (x))

= i \frac{1}{2} (arcsin^{2} (x) + Li_{2} (e^{2 i a r c s i n (x)}) - arcsin (x) ln (1 - (- 1)^{\frac{2}{π} a r c s i n (x)}) + arcsin (x) ln (x))

Füge eine Konstante zur Lösung hinzu

= i \frac{1}{2} (arcsin^{2} (x) + Li_{2} (e^{2 i a r c s i n (x)}) - arcsin (x) ln (1 - (- 1)^{\frac{2}{π} a r c s i n (x)}) + arcsin (x) ln (x)) + C

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