integral of sin^{12}(x)cos^7(x)

Lösung

\int sin^{12} (x) cos^{7} (x) d x

\frac{sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{5} + \frac{3 sin ^{17} ( x )}{17} - \frac{sin ^{19} ( x )}{19} + C

Schritte zur Lösung

\int sin^{12} (x) cos^{7} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (x))^{3} cos (x) sin^{12} (x) d x

Wende U-Substitution an

= \int u^{12} (1 - u^{2})^{3} d u

Multipliziere aus

u^{12} (1 - u^{2})^{3} : u^{12} - 3 u^{14} + 3 u^{16} - u^{18}

= \int u^{12} - 3 u^{14} + 3 u^{16} - u^{18} 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^{12} d u - \int 3 u^{14} d u + \int 3 u^{16} d u - \int u^{18} d u

\int u^{12} d u = \frac{u ^{13}}{13}

\int 3 u^{14} d u = \frac{u ^{15}}{5}

\int 3 u^{16} d u = \frac{3 u ^{17}}{17}

\int u^{18} d u = \frac{u ^{19}}{19}

= \frac{u ^{13}}{13} - \frac{u ^{15}}{5} + \frac{3 u ^{17}}{17} - \frac{u ^{19}}{19}

Setze in

u = sin (x)

ein

= \frac{sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{5} + \frac{3 sin ^{17} ( x )}{17} - \frac{sin ^{19} ( x )}{19}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{5} + \frac{3 sin ^{17} ( x )}{17} - \frac{sin ^{19} ( x )}{19} + C