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Beliebt Rechnen >

integral of sin^6(x)cos^5(x)

Lösung

\int sin^{6} (x) cos^{5} (x) d x

Lösung

\frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

Schritte zur Lösung

\int sin^{6} (x) cos^{5} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int u^{6} - 2 u^{8} + u^{10} 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 2 u^{8} d u + \int u^{10} d u

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

\int 2 u^{8} d u = \frac{2 u ^{9}}{9}

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

= \frac{u ^{7}}{7} - \frac{2 u ^{9}}{9} + \frac{u ^{11}}{11}

Setze in

u = sin (x)

ein

= \frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

Graph

Beliebte Beispiele

integral of e^x+x^e+e+x

\int e^{x} + x^{e} + e + x d x

derivative f(x)=sin(4x)-cos(3x)

d er i v a t i v e f (x) = sin (4 x) - cos (3 x)

(dy)/(dx)=x^3+x^2-5x+11

\frac{d y}{d x} = x^{3} + x^{2} - 5 x + 11

integral from 1 to 1000 of (21)/(x^3)

\int_{1}^{1000} \frac{21}{x ^{3}} d x

y^{''}-2y^'+5y=e^tcos(2t)

y^{^{''}} - 2 y^{^{'}} + 5 y = e^{t} cos (2 t)