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Beliebt Rechnen >

integral of cos^5(x)tan^3(x)

Lösung

\int cos^{5} (x) tan^{3} (x) d x

Lösung

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

Schritte zur Lösung

\int cos^{5} (x) tan^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= \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

= - \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}

= - \frac{u ^{3}}{3} + \frac{u ^{5}}{5}

Setze in

u = cos (x)

ein

= - \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5}

Füge eine Konstante zur Lösung hinzu

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

Graph

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