integral of (sin(x))^3(cos(x))^5

Lösung

\int (sin (x))^{3} (cos (x))^{5} d x

- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8} + C

Schritte zur Lösung

\int (sin (x))^{3} (cos (x))^{5} d x

Vereinfache

= \int cos^{5} (x) sin^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x)) sin (x) cos^{5} (x) d x

Wende U-Substitution an

= \int - u^{5} (1 - u^{2}) d u

Multipliziere aus

- u^{5} (1 - u^{2}) : - u^{5} + u^{7}

= \int - u^{5} + u^{7} 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^{5} d u + \int u^{7} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int u^{7} d u = \frac{u ^{8}}{8}

= - \frac{u ^{6}}{6} + \frac{u ^{8}}{8}

Setze in

u = cos (x)

ein

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8} + C

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