integral of sin^5(x)*cos^2(x)

Lösung

\int sin^{5} (x) \cdot cos^{2} (x) d x

- \frac{cos ^{3} ( x )}{3} + \frac{2 cos ^{5} ( x )}{5} - \frac{cos ^{7} ( x )}{7} + C

Schritte zur Lösung

\int sin^{5} (x) cos^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

- u^{2} (1 - u^{2})^{2} : - u^{2} + 2 u^{4} - u^{6}

= \int - u^{2} + 2 u^{4} - u^{6} 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 2 u^{4} d u - \int u^{6} d u

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

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= - \frac{u ^{3}}{3} + \frac{2 u ^{5}}{5} - \frac{u ^{7}}{7}

Setze in

u = cos (x)

ein

= - \frac{cos ^{3} ( x )}{3} + \frac{2 cos ^{5} ( x )}{5} - \frac{cos ^{7} ( x )}{7}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{3} ( x )}{3} + \frac{2 cos ^{5} ( x )}{5} - \frac{cos ^{7} ( x )}{7} + C