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Beliebt Rechnen >

integral of (sin(x))^4(cos(x))^3

Lösung

\int (sin (x))^{4} (cos (x))^{3} d x

Lösung

\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7} + C

Schritte zur Lösung

\int (sin (x))^{4} (cos (x))^{3} d x

Wende U-Substitution an

= \int u^{4} (1 - u^{2}) d u

Multipliziere aus

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

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

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

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

= \frac{u ^{5}}{5} - \frac{u ^{7}}{7}

Setze in

u = sin (x)

ein

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7} + C

Graph

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