integral of sin^{100}(x)cos^3(x)

Lösung

\int sin^{100} (x) cos^{3} (x) d x

\frac{sin ^{101} ( x )}{101} - \frac{sin ^{103} ( x )}{103} + C

Schritte zur Lösung

\int sin^{100} (x) cos^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (x)) cos (x) sin^{100} (x) d x

Wende U-Substitution an

= \int u^{100} (1 - u^{2}) d u

Multipliziere aus

u^{100} (1 - u^{2}) : u^{100} - u^{102}

= \int u^{100} - u^{102} 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^{100} d u - \int u^{102} d u

\int u^{100} d u = \frac{u ^{101}}{101}

\int u^{102} d u = \frac{u ^{103}}{103}

= \frac{u ^{101}}{101} - \frac{u ^{103}}{103}

Setze in

u = sin (x)

ein

= \frac{sin ^{101} ( x )}{101} - \frac{sin ^{103} ( x )}{103}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{101} ( x )}{101} - \frac{sin ^{103} ( x )}{103} + C

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