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Beliebt Rechnen >

integral of cos^3(x)ln(sin(x))

Lösung

\int cos^{3} (x) ln (sin (x)) d x

Lösung

- \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x) + C

Schritte zur Lösung

\int cos^{3} (x) ln (sin (x)) d x

Trigonometrische Substitution anwenden

= \int ln (u) (- u^{2} + 1) d u

Wende die partielle Integration an

= ln (u) (- \frac{u ^{3}}{3} + u) - \int \frac{1}{3} (- u^{2} + 3) d u

\int \frac{1}{3} (- u^{2} + 3) d u = - \frac{u ^{3}}{9} + u

= ln (u) (- \frac{u ^{3}}{3} + u) - (- \frac{u ^{3}}{9} + u)

Setze in

u = sin (x)

ein

= ln (sin (x)) (- \frac{sin ^{3} ( x )}{3} + sin (x)) - (- \frac{sin ^{3} ( x )}{9} + sin (x))

Vereinfache

ln (sin (x)) (- \frac{sin ^{3} ( x )}{3} + sin (x)) - (- \frac{sin ^{3} ( x )}{9} + sin (x)) : - \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x)

= - \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x) + C

Graph

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