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Beliebt Rechnen >

integral of sin^{10}(θ)cos^3(θ)

Lösung

\int sin^{10} (θ) cos^{3} (θ) d θ

Lösung

\frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13} + C

Schritte zur Lösung

\int sin^{10} (θ) cos^{3} (θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (θ)) cos (θ) sin^{10} (θ) d θ

Wende U-Substitution an

= \int u^{10} (1 - u^{2}) d u

Multipliziere aus

u^{10} (1 - u^{2}) : u^{10} - u^{12}

= \int u^{10} - u^{12} 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^{10} d u - \int u^{12} d u

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

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

= \frac{u ^{11}}{11} - \frac{u ^{13}}{13}

Setze in

u = sin (θ)

ein

= \frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13} + C

Graph

Beliebte Beispiele

integral of ((sqrt(x)-2)^2)/(x^3)

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\frac{d}{d x} (x^{4} - 2 x^{2} - 3)

integral of (x+7)/(x^2+25)

\int \frac{x + 7}{x ^{2} + 25} d x

limit as x approaches 0 of (1+x)/x

x \to 0 lim (\frac{1 + x}{x})