integral of 3sin^4(xco)s^2x

Lösung

\int 3 sin^{4} (x co) s^{2} x d x

\frac{3 s ^{2}}{32 c ^{2} o ^{2}} (6 c^{2} o^{2} x^{2} - 8 co x sin^{3} (co x) cos (co x) - 6 co x sin (2 co x) + 2 sin^{4} (co x) - 3 cos (2 co x)) + C

Schritte zur Lösung

\int 3 sin^{4} (x co) s^{2} x d x

Entferne die Konstante:

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

= 3 s^{2} \cdot \int sin^{4} (co x) x d x

Wende U-Substitution an

= 3 s^{2} \cdot \int \frac{u sin ^{4} ( 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^{2} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{4} (u) d u

Wende die partielle Integration an

= 3 s^{2} \frac{1}{c ^{2} o ^{2}} (u (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))) - \int - \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) d u)

\int - \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) d u = - \frac{1}{16} sin^{4} (u) + \frac{3}{8} (\frac{u ^{2}}{2} + \frac{1}{4} cos (2 u))

= 3 s^{2} \frac{1}{c ^{2} o ^{2}} (u (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))) - (- \frac{1}{16} sin^{4} (u) + \frac{3}{8} (\frac{u ^{2}}{2} + \frac{1}{4} cos (2 u))))

Setze in

u = co x

ein

= 3 s^{2} \frac{1}{c ^{2} o ^{2}} (co x (- \frac{1}{4} sin^{3} (co x) cos (co x) + \frac{3}{8} (co x - \frac{1}{2} sin (2 co x))) - (- \frac{1}{16} sin^{4} (co x) + \frac{3}{8} (\frac{( co x ) ^{2}}{2} + \frac{1}{4} cos (2 co x))))

Vereinfache

3 s^{2} \frac{1}{c ^{2} o ^{2}} (co x (- \frac{1}{4} sin^{3} (co x) cos (co x) + \frac{3}{8} (co x - \frac{1}{2} sin (2 co x))) - (- \frac{1}{16} sin^{4} (co x) + \frac{3}{8} (\frac{( co x ) ^{2}}{2} + \frac{1}{4} cos (2 co x)))) : \frac{3 s ^{2}}{32 c ^{2} o ^{2}} (6 c^{2} o^{2} x^{2} - 8 co x sin^{3} (co x) cos (co x) - 6 co x sin (2 co x) + 2 sin^{4} (co x) - 3 cos (2 co x))

= \frac{3 s ^{2}}{32 c ^{2} o ^{2}} (6 c^{2} o^{2} x^{2} - 8 co x sin^{3} (co x) cos (co x) - 6 co x sin (2 co x) + 2 sin^{4} (co x) - 3 cos (2 co x))

Füge eine Konstante zur Lösung hinzu

= \frac{3 s ^{2}}{32 c ^{2} o ^{2}} (6 c^{2} o^{2} x^{2} - 8 co x sin^{3} (co x) cos (co x) - 6 co x sin (2 co x) + 2 sin^{4} (co x) - 3 cos (2 co x)) + C

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