integral of cos^7(4x)sin^4(4x)

Lösung

\int cos^{7} (4 x) sin^{4} (4 x) d x

\frac{1}{4} (\frac{sin ^{5} ( 4 x )}{5} - \frac{3 sin ^{7} ( 4 x )}{7} + \frac{sin ^{9} ( 4 x )}{3} - \frac{sin ^{11} ( 4 x )}{11}) + C

Schritte zur Lösung

\int cos^{7} (4 x) sin^{4} (4 x) d x

Wende U-Substitution an

= \int cos^{7} (u) sin^{4} (u) \frac{1}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int cos^{7} (u) sin^{4} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

v^{4} (1 - v^{2})^{3} : v^{4} - 3 v^{6} + 3 v^{8} - v^{10}

= \frac{1}{4} \cdot \int v^{4} - 3 v^{6} + 3 v^{8} - v^{10} d v

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}{4} (\int v^{4} d v - \int 3 v^{6} d v + \int 3 v^{8} d v - \int v^{10} d v)

\int v^{4} d v = \frac{v ^{5}}{5}

\int 3 v^{6} d v = \frac{3 v ^{7}}{7}

\int 3 v^{8} d v = \frac{v ^{9}}{3}

\int v^{10} d v = \frac{v ^{11}}{11}

= \frac{1}{4} (\frac{v ^{5}}{5} - \frac{3 v ^{7}}{7} + \frac{v ^{9}}{3} - \frac{v ^{11}}{11})

Ersetze zurück

= \frac{1}{4} (\frac{sin ^{5} ( 4 x )}{5} - \frac{3 sin ^{7} ( 4 x )}{7} + \frac{sin ^{9} ( 4 x )}{3} - \frac{sin ^{11} ( 4 x )}{11})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (\frac{sin ^{5} ( 4 x )}{5} - \frac{3 sin ^{7} ( 4 x )}{7} + \frac{sin ^{9} ( 4 x )}{3} - \frac{sin ^{11} ( 4 x )}{11}) + C

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