de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of sin^7(x)cos^6(x)

Lösung

\int sin^{7} (x) cos^{6} (x) d x

Lösung

- \frac{cos ^{7} ( x )}{7} + \frac{cos ^{9} ( x )}{3} - \frac{3 cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13} + C

Schritte zur Lösung

\int sin^{7} (x) cos^{6} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{3} sin (x) cos^{6} (x) d x

Wende U-Substitution an

= \int - u^{6} (1 - u^{2})^{3} d u

Multipliziere aus

- u^{6} (1 - u^{2})^{3} : - u^{6} + 3 u^{8} - 3 u^{10} + u^{12}

= \int - u^{6} + 3 u^{8} - 3 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^{6} d u + \int 3 u^{8} d u - \int 3 u^{10} d u + \int u^{12} d u

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

\int 3 u^{8} d u = \frac{u ^{9}}{3}

\int 3 u^{10} d u = \frac{3 u ^{11}}{11}

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

= - \frac{u ^{7}}{7} + \frac{u ^{9}}{3} - \frac{3 u ^{11}}{11} + \frac{u ^{13}}{13}

Setze in

u = cos (x)

ein

= - \frac{cos ^{7} ( x )}{7} + \frac{cos ^{9} ( x )}{3} - \frac{3 cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{7} ( x )}{7} + \frac{cos ^{9} ( x )}{3} - \frac{3 cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13} + C

Graph

Beliebte Beispiele

integral of (x+1)/(x^2+x-6)

integral of (e^{2x}-3e^x)/(e^x+1)

integral of (x^2-3x+2)cos(x)

integral of x^3(x^4-5)^4

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