integral of sin^3(xco)s^{-2}x

Lösung

\int sin^{3} (x co) s^{- 2} x d x

\frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x)) + C

Schritte zur Lösung

\int sin^{3} (x co) s^{- 2} x d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= s^{- 2} \cdot \int sin^{3} (co x) x d x

Wende Exponentenregel an:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{s ^{2}} \cdot \int sin^{3} (co x) x d x

Wende U-Substitution an

= \frac{1}{s ^{2}} \cdot \int \frac{u sin ^{3} ( u )}{c ^{2} o ^{2}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

Wende die partielle Integration an

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - \int - cos (u) + \frac{1}{3} cos^{3} (u) d u)

\int - cos (u) + \frac{1}{3} cos^{3} (u) d u = - \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - (- \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)))

Setze in

u = co x

ein

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x)))

Vereinfache

\frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x))) : \frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x))

= \frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x)) + C

d er i v a t i v e 7 - x

d er i v a t i v e sin^{5} (x)

\int \frac{x + 3}{4 - ( x - 2 ) ^{2}} d x

d er i v a t i v e 2^{90}

d er i v a t i v e f (x) = 2 x^{2} - 7 x + 5