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Beliebt Rechnen >

integral of cot^3(a)csc^3(a)

Lösung

\int cot^{3} (a) csc^{3} (a) d a

Lösung

- \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a) + C

Schritte zur Lösung

\int cot^{3} (a) csc^{3} (a) d a

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

u^{2} (u^{2} - 1) : u^{4} - u^{2}

= - \int u^{4} - u^{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^{4} d u - \int u^{2} d u)

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

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

= - (\frac{u ^{5}}{5} - \frac{u ^{3}}{3})

Setze in

u = csc (a)

ein

= - (\frac{csc ^{5} ( a )}{5} - \frac{csc ^{3} ( a )}{3})

Vereinfache

- (\frac{csc ^{5} ( a )}{5} - \frac{csc ^{3} ( a )}{3}) : - \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a)

= - \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a) + C

Graph

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