integral of 10sin^3(xco)s^5x

Lösung

\int 10 sin^{3} (x co) s^{5} x d x

s^{5} \frac{10}{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 10 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

= 10 s^{5} \cdot \int sin^{3} (co x) x d x

Wende U-Substitution an

= 10 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

= 10 s^{5} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

Wende die partielle Integration an

= 10 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)

= 10 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

= 10 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

10 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))) : s^{5} \frac{10}{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^{5} \frac{10}{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^{5} \frac{10}{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

in v erse l a pl a ce \frac{1}{s ^{2} - 9}

d er i v a t i v e (2 x - 4)^{4} (x^{2} + x + 1)^{5}

\int \frac{6 x ^{2}}{x ^{3} - 1} d x

in v erse l a pl a ce 3 s

d er i v a t i v e y = 7 x + 5