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Beliebt Rechnen >

integral of cot^5(3-2x)csc^3(3-2x)

Lösung

\int cot^{5} (3 - 2 x) csc^{3} (3 - 2 x) d x

Lösung

\frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x) + C

Schritte zur Lösung

\int cot^{5} (3 - 2 x) csc^{3} (3 - 2 x) d x

Wende U-Substitution an

= \int - \frac{1}{2} cot^{5} (u) csc^{3} (u) d u

Entferne die Konstante:

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

= - \frac{1}{2} \cdot \int cot^{5} (u) csc^{3} (u) d u

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

v^{2} (v^{2} - 1)^{2} : v^{6} - 2 v^{4} + v^{2}

= - \frac{1}{2} (- \int v^{6} - 2 v^{4} + v^{2} d v)

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{2} (- (\int v^{6} d v - \int 2 v^{4} d v + \int v^{2} d v))

\int v^{6} d v = \frac{v ^{7}}{7}

\int 2 v^{4} d v = \frac{2 v ^{5}}{5}

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

= - \frac{1}{2} (- (\frac{v ^{7}}{7} - \frac{2 v ^{5}}{5} + \frac{v ^{3}}{3}))

Ersetze zurück

= - \frac{1}{2} (- (\frac{csc ^{7} ( 3 - 2 x )}{7} - \frac{2 csc ^{5} ( 3 - 2 x )}{5} + \frac{csc ^{3} ( 3 - 2 x )}{3}))

Vereinfache

- \frac{1}{2} (- (\frac{csc ^{7} ( 3 - 2 x )}{7} - \frac{2 csc ^{5} ( 3 - 2 x )}{5} + \frac{csc ^{3} ( 3 - 2 x )}{3})) : \frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x)

= \frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x) + C

Graph

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