integral of sin^4(3x)cos^4(3x)

Lösung

\int sin^{4} (3 x) cos^{4} (3 x) d x

\frac{1}{768} (18 x - 4 sin^{3} (3 x) cos (3 x) - 3 sin (6 x) + 48 sin^{5} (3 x) cos (3 x) - 32 sin^{7} (3 x) cos (3 x)) + C

Schritte zur Lösung

\int sin^{4} (3 x) cos^{4} (3 x) d x

Wende U-Substitution an

= \int sin^{4} (u) cos^{4} (u) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int sin^{4} (u) cos^{4} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

sin^{4} (u) (1 - sin^{2} (u))^{2} : sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u)

= \frac{1}{3} \cdot \int sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int sin^{4} (u) d u - \int 2 sin^{6} (u) d u + \int sin^{8} (u) d u)

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

\int 2 sin^{6} (u) d u = 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

\int sin^{8} (u) d u = - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

= \frac{1}{3} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) - 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))) - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))))

Setze in

u = 3 x

ein

= \frac{1}{3} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)) - 2 (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)))) - \frac{cos ( 3 x ) sin ^{7} ( 3 x )}{8} + \frac{7}{8} (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)))))

Vereinfache

\frac{1}{3} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)) - 2 (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)))) - \frac{cos ( 3 x ) sin ^{7} ( 3 x )}{8} + \frac{7}{8} (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x))))) : \frac{1}{768} (18 x - 4 sin^{3} (3 x) cos (3 x) - 3 sin (6 x) + 48 sin^{5} (3 x) cos (3 x) - 32 sin^{7} (3 x) cos (3 x))

= \frac{1}{768} (18 x - 4 sin^{3} (3 x) cos (3 x) - 3 sin (6 x) + 48 sin^{5} (3 x) cos (3 x) - 32 sin^{7} (3 x) cos (3 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{768} (18 x - 4 sin^{3} (3 x) cos (3 x) - 3 sin (6 x) + 48 sin^{5} (3 x) cos (3 x) - 32 sin^{7} (3 x) cos (3 x)) + C

t an g e n t f (x) = 9 + 5 x + 2 x e^{x}, at x = 0

\frac{\partial}{\partial y} (2 x y^{4} e^{y} + 2 x y^{3} + y)

t an g e n t f (x) = 19 - x^{2}, at x = 3

x y^{^{'}} - 2 y^{^{'}} - (x - 2) sin (x) + y = 0

\frac{d}{d x} (\frac{- y}{x})