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

Lösung

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

- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10} + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

- u^{5} (1 - u^{2})^{2} : - u^{5} + 2 u^{7} - u^{9}

= \int - u^{5} + 2 u^{7} - u^{9} 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 2 u^{7} d u - \int u^{9} d u

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

\int 2 u^{7} d u = \frac{u ^{8}}{4}

\int u^{9} d u = \frac{u ^{10}}{10}

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

Setze in

u = cos (x)

ein

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10} + C

x \to - 9 - lim (x^{2} - 81)

\int 2 x^{2} cos (m x) d x

x \to - 3 lim (4 x^{2} + 2 x - 1)

\int sin^{6} (π x) d x

n = 7 \sum \infty \frac{1}{6 ^{n}}