integral of 9sin^3(xco)s^5x

Lösung

\int 9 sin^{3} (x co) s^{5} x d x

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

Schritte zur Lösung

\int 9 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

= 9 s^{5} \cdot \int sin^{3} (co x) x d x

Wende U-Substitution an

= 9 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

= 9 s^{5} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

Wende die partielle Integration an

= 9 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)

= 9 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

= 9 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

9 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{9 s ^{5}}{co} x cos (co x) + \frac{3 s ^{5}}{co} x cos^{3} (co x) + \frac{6 s ^{5}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{5}}{c ^{2} o ^{2}} sin^{3} (co x)

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

Füge eine Konstante zur Lösung hinzu

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

\int - \frac{1}{sec ^{2} ( x )} d x

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

t an g e n t f (x) = 12 x - 1.86 x^{2}, at x = a

d er i v a t i v e y = (3 x + 1) 8 x - 1

\frac{d y}{d x} = x y