de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (cos^5(t))/(sqrt(sin(t)))

Lösung

\int \frac{cos ^{5} ( t )}{sin ( t )} d t

Lösung

2 sin (t) - \frac{4}{5} sin^{\frac{5}{2}} (t) + \frac{2}{9} sin^{\frac{9}{2}} (t) + C

Schritte zur Lösung

\int \frac{cos ^{5} ( t )}{sin ( t )} d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( 1 - u ^{2} ) ^{2}}{u} : \frac{1}{u} - 2 u^{\frac{3}{2}} + u^{\frac{7}{2}}

= \int \frac{1}{u} - 2 u^{\frac{3}{2}} + u^{\frac{7}{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 \frac{1}{u} d u - \int 2 u^{\frac{3}{2}} d u + \int u^{\frac{7}{2}} d u

\int \frac{1}{u} d u = 2 u

\int 2 u^{\frac{3}{2}} d u = \frac{4}{5} u^{\frac{5}{2}}

\int u^{\frac{7}{2}} d u = \frac{2}{9} u^{\frac{9}{2}}

= 2 u - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{9} u^{\frac{9}{2}}

Setze in

u = sin (t)

ein

= 2 sin (t) - \frac{4}{5} sin^{\frac{5}{2}} (t) + \frac{2}{9} sin^{\frac{9}{2}} (t)

Füge eine Konstante zur Lösung hinzu

= 2 sin (t) - \frac{4}{5} sin^{\frac{5}{2}} (t) + \frac{2}{9} sin^{\frac{9}{2}} (t) + C

Graph

Beliebte Beispiele

derivative of sqrt(10+\sqrt{3x)}

\frac{d}{d x} (10 + 3 x)

derivative of y^2-xsin(y-xye^{-y})

\frac{d}{d x} (y^{2} - x sin (y) - x y e^{- y})

integral of sqrt((1+x^2))

\int (1 + x^{2}) d x

derivative of x/(3-{z(x)})

\frac{d}{d x} (\frac{x}{3 - z ( x )})

slopeintercept (3)(7.8)

s l o p e in t erce pt (3) (7.8)