integral of cos^6(x/2)sin^4(x/2)

Lösung

\int cos^{6} (\frac{x}{2}) sin^{4} (\frac{x}{2}) d x

2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2})) + C

Schritte zur Lösung

\int cos^{6} (\frac{x}{2}) sin^{4} (\frac{x}{2}) d x

Wende U-Substitution an

= \int cos^{6} (u) sin^{4} (u) \cdot 2 d u

Entferne die Konstante:

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

= 2 \cdot \int cos^{6} (u) sin^{4} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \cdot \int cos^{6} (u) (1 - cos^{2} (u))^{2} d u

Multipliziere aus

cos^{6} (u) (1 - cos^{2} (u))^{2} : cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (u)

= 2 \cdot \int cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (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

= 2 (\int cos^{6} (u) d u - \int 2 cos^{8} (u) d u + \int cos^{10} (u) d u)

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

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

\int cos^{10} (u) d u = \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

= 2 (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))) - 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))) + \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))))

Setze in

u = \frac{x}{2}

ein

= 2 (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))) - 2 (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))))) + \frac{sin ( \frac{x}{2} ) cos ^{9} ( \frac{x}{2} )}{10} + \frac{9}{10} (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))))))

Vereinfache

2 (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))) - 2 (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))))) + \frac{sin ( \frac{x}{2} ) cos ^{9} ( \frac{x}{2} )}{10} + \frac{9}{10} (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2})))))) : 2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2}))

= 2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2})) + C

\int \frac{sin ( θ )}{cos ^{2} ( θ ) + cos ( θ ) - 2} d θ

\int 32 arctan (x) d x

\int x^{2} 3 x^{3} - 2 d x

\int (x^{3} - 7 x) \cdot e^{5 x} d x

\int \frac{sin ( π x )}{x ^{2}} d x