integral of (sin(x))^6(cos(x))^3

Lösung

\int (sin (x))^{6} (cos (x))^{3} d x

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

Schritte zur Lösung

\int (sin (x))^{6} (cos (x))^{3} d x

Vereinfache

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int u^{6} - u^{8} 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 u^{8} d u

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

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

= \frac{u ^{7}}{7} - \frac{u ^{9}}{9}

Setze in

u = sin (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

\int sec^{6} (4 x) d x

t an g e n t f (x) = (3 x^{3} - 4 x^{2} + 5 x - 9)^{2}, at x = 2

x \to 2 lim (x^{2} + 6 x + 5)

x \to 0 + lim (x^{13 x})

x \to 4 lim (6)