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

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int - u^{4} + 3 u^{6} - 3 u^{8} + u^{10} 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

= - \int u^{4} d u + \int 3 u^{6} d u - \int 3 u^{8} d u + \int u^{10} d u

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

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

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

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

= - \frac{u ^{5}}{5} + \frac{3 u ^{7}}{7} - \frac{u ^{9}}{3} + \frac{u ^{11}}{11}

Setze in

u = cos (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

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