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

Lösung

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

- (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x))) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

\int - 2 u sin (u) d u = - 2 (- u cos (u) + sin (u))

= - (sin (u) (1 - u^{2}) - (- 2 (- u cos (u) + sin (u))))

Setze in

u = cos (x)

ein

= - (sin (cos (x)) (1 - cos^{2} (x)) - (- 2 (- cos (x) cos (cos (x)) + sin (cos (x)))))

Vereinfache

- (sin (cos (x)) (1 - cos^{2} (x)) - (- 2 (- cos (x) cos (cos (x)) + sin (cos (x))))) : - (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x)))

= - (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x)))

Füge eine Konstante zur Lösung hinzu

= - (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x))) + C

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