integral of sin^5(xco)s^6x

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

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 x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (x) = \frac{x ^{3} - 7 x ^{2} + 7}{x ^{2}}

\frac{\partial}{\partial x} (y^{6} sin (2 x))

x \to \infty lim (- 3 x)

\frac{d}{d x} (a + \frac{1}{x})

a re a 1 - x^{2}, x^{2}