integral of 5sin^3(xco)s^7x

Lösung

\int 5 sin^{3} (x co) s^{7} x d x

s^{7} \frac{5}{c ^{2} o ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x)) + C

Schritte zur Lösung

\int 5 sin^{3} (x co) s^{7} x d x

Entferne die Konstante:

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

= 5 s^{7} \cdot \int sin^{3} (co x) x d x

Wende U-Substitution an

= 5 s^{7} \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

= 5 s^{7} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

Wende die partielle Integration an

= 5 s^{7} \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)

= 5 s^{7} \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

= 5 s^{7} \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

5 s^{7} \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))) : s^{7} \frac{5}{c ^{2} o ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x))

= s^{7} \frac{5}{c ^{2} o ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x))

Füge eine Konstante zur Lösung hinzu

= s^{7} \frac{5}{c ^{2} o ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x)) + C

\int (5 x^{4} + \frac{x}{7}) d x

\frac{d y}{d x} + y + 2 cos (x) = 0

\frac{d}{d x} (\frac{( x ^{2} + 3 )}{x})

in v erse l a pl a ce \frac{s}{s ^{2} + 9}

\frac{d}{d x} (\frac{2 x - 9}{x})