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

Lösung

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

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

u (1 - u^{2}) : u - u^{\frac{5}{2}}

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

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

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

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

Setze in

u = sin (x)

ein

= \frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x)

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (x) = 5 x^{4} - \frac{2}{x}

d er i v a t i v e y = \frac{a ^{4}}{a ^{3} - 8 x}

\frac{\partial}{\partial z} (z y^{2} (t + 1))

\int \frac{sec ( y )}{sin ( y )} d y

\frac{\partial}{\partial x} (\frac{1}{x ^{2}})