integral of sin^5(tco)s^6t

Lösung

\int sin^{5} (t co) s^{6} t d t

\frac{s ^{6}}{225 c ^{2} o ^{2}} (- 225 co t cos (co t) + 150 co t cos^{3} (co t) - 45 co t cos^{5} (co t) + 120 sin (co t) + 20 sin^{3} (co t) + 9 sin^{5} (co t)) + C

Schritte zur Lösung

\int sin^{5} (t co) s^{6} t d t

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= s^{6} \cdot \int sin^{5} (co t) t d t

Wende U-Substitution an

= s^{6} \cdot \int \frac{u sin ^{5} ( u )}{c ^{2} o ^{2}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= s^{6} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{5} (u) d u

Wende die partielle Integration an

= s^{6} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{2}{3} cos^{3} (u) - \frac{1}{5} cos^{5} (u)) - \int - cos (u) + \frac{2}{3} cos^{3} (u) - \frac{1}{5} cos^{5} (u) d u)

\int - cos (u) + \frac{2}{3} cos^{3} (u) - \frac{1}{5} cos^{5} (u) d u = - \frac{8}{15} sin (u) - \frac{2}{9} sin^{3} (u) + \frac{2}{15} sin^{3} (u) - \frac{1}{25} sin^{5} (u)

= s^{6} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{2}{3} cos^{3} (u) - \frac{1}{5} cos^{5} (u)) - (- \frac{8}{15} sin (u) - \frac{2}{9} sin^{3} (u) + \frac{2}{15} sin^{3} (u) - \frac{1}{25} sin^{5} (u)))

Setze in

u = co t

ein

= s^{6} \frac{1}{c ^{2} o ^{2}} (co t (- cos (co t) + \frac{2}{3} cos^{3} (co t) - \frac{1}{5} cos^{5} (co t)) - (- \frac{8}{15} sin (co t) - \frac{2}{9} sin^{3} (co t) + \frac{2}{15} sin^{3} (co t) - \frac{1}{25} sin^{5} (co t)))

Vereinfache

s^{6} \frac{1}{c ^{2} o ^{2}} (co t (- cos (co t) + \frac{2}{3} cos^{3} (co t) - \frac{1}{5} cos^{5} (co t)) - (- \frac{8}{15} sin (co t) - \frac{2}{9} sin^{3} (co t) + \frac{2}{15} sin^{3} (co t) - \frac{1}{25} sin^{5} (co t))) : \frac{s ^{6}}{225 c ^{2} o ^{2}} (- 225 co t cos (co t) + 150 co t cos^{3} (co t) - 45 co t cos^{5} (co t) + 120 sin (co t) + 20 sin^{3} (co t) + 9 sin^{5} (co t))

= \frac{s ^{6}}{225 c ^{2} o ^{2}} (- 225 co t cos (co t) + 150 co t cos^{3} (co t) - 45 co t cos^{5} (co t) + 120 sin (co t) + 20 sin^{3} (co t) + 9 sin^{5} (co t))

Füge eine Konstante zur Lösung hinzu

= \frac{s ^{6}}{225 c ^{2} o ^{2}} (- 225 co t cos (co t) + 150 co t cos^{3} (co t) - 45 co t cos^{5} (co t) + 120 sin (co t) + 20 sin^{3} (co t) + 9 sin^{5} (co t)) + C

x^{^{'}} = - \frac{x}{x + 1}

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