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Beliebt Rechnen >

integral of 3sin^3(xco)s^5x

Lösung

\int 3 sin^{3} (x co) s^{5} x d x

Lösung

- \frac{3 s ^{5}}{co} x cos (co x) + \frac{s ^{5}}{co} x cos^{3} (co x) + \frac{2 s ^{5}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{5}}{3 c ^{2} o ^{2}} sin^{3} (co x) + C

Schritte zur Lösung

\int 3 sin^{3} (x co) s^{5} x d x

Entferne die Konstante:

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

= 3 s^{5} \cdot \int sin^{3} (co x) x d x

Wende U-Substitution an

= 3 s^{5} \cdot \int \frac{u sin ^{3} ( u )}{c ^{2} o ^{2}} d u

Entferne die Konstante:

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

= 3 s^{5} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

Wende die partielle Integration an

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

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

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

Setze in

u = co x

ein

= 3 s^{5} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x)))

Vereinfache

3 s^{5} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x))) : - \frac{3 s ^{5}}{co} x cos (co x) + \frac{s ^{5}}{co} x cos^{3} (co x) + \frac{2 s ^{5}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{5}}{3 c ^{2} o ^{2}} sin^{3} (co x)

= - \frac{3 s ^{5}}{co} x cos (co x) + \frac{s ^{5}}{co} x cos^{3} (co x) + \frac{2 s ^{5}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{5}}{3 c ^{2} o ^{2}} sin^{3} (co x)

Füge eine Konstante zur Lösung hinzu

= - \frac{3 s ^{5}}{co} x cos (co x) + \frac{s ^{5}}{co} x cos^{3} (co x) + \frac{2 s ^{5}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{5}}{3 c ^{2} o ^{2}} sin^{3} (co x) + C

Graph

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