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

Lösung

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

- \frac{1}{2} csc^{2} (x) - ln ∣ sin (x) ∣ + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

\frac{1 - u ^{2}}{u ^{3}} : \frac{1}{u ^{3}} - \frac{1}{u}

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

= - \frac{1}{2 u ^{2}} - ln ∣ u ∣

Setze in

u = sin (x)

ein

= - \frac{1}{2 sin ^{2} ( x )} - ln ∣ sin (x) ∣

Vereinfache

- \frac{1}{2 sin ^{2} ( x )} - ln ∣ sin (x) ∣ : - \frac{1}{2} csc^{2} (x) - ln ∣ sin (x) ∣

= - \frac{1}{2} csc^{2} (x) - ln ∣ sin (x) ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} csc^{2} (x) - ln ∣ sin (x) ∣ + C

\int x 25 x - 8 d x

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