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Beliebt Rechnen >

integral of 18sin^2(x)cos^3(x)

Lösung

\int 18 sin^{2} (x) cos^{3} (x) d x

Lösung

18 (\frac{sin ^{3} ( x )}{3} - \frac{sin ^{5} ( x )}{5}) + C

Schritte zur Lösung

\int 18 sin^{2} (x) cos^{3} (x) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 18 \cdot \int sin^{2} (x) cos^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 18 \cdot \int (1 - sin^{2} (x)) cos (x) sin^{2} (x) d x

Wende U-Substitution an

= 18 \cdot \int u^{2} (1 - u^{2}) d u

Multipliziere aus

u^{2} (1 - u^{2}) : u^{2} - u^{4}

= 18 \cdot \int u^{2} - u^{4} 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

= 18 (\int u^{2} d u - \int u^{4} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= 18 (\frac{u ^{3}}{3} - \frac{u ^{5}}{5})

Setze in

u = sin (x)

ein

= 18 (\frac{sin ^{3} ( x )}{3} - \frac{sin ^{5} ( x )}{5})

Füge eine Konstante zur Lösung hinzu

= 18 (\frac{sin ^{3} ( x )}{3} - \frac{sin ^{5} ( x )}{5}) + C

Graph

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