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Beliebt Rechnen >

integral of (cot^5(β))/(sqrt(sin(β)))

Lösung

\int \frac{cot ^{5} ( β )}{sin ( β )} d β

Lösung

- \frac{2}{9} csc^{\frac{9}{2}} (β) + \frac{4}{5} csc^{\frac{5}{2}} (β) - 2 csc (β) + C

Schritte zur Lösung

\int \frac{cot ^{5} ( β )}{sin ( β )} d β

Wende U-Substitution an

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

Multipliziere aus

\frac{( 1 - u ^{2} ) ^{2}}{u ^{5} u} : \frac{1}{u ^{5} u} - \frac{2}{u ^{3} u} + \frac{1}{u u}

= \int \frac{1}{u ^{5} u} - \frac{2}{u ^{3} u} + \frac{1}{u 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 ^{5} u} d u - \int \frac{2}{u ^{3} u} d u + \int \frac{1}{u u} d u

\int \frac{1}{u ^{5} u} d u = - \frac{2}{9 u ^{\frac{9}{2}}}

\int \frac{2}{u ^{3} u} d u = - \frac{4}{5 u ^{\frac{5}{2}}}

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

= - \frac{2}{9 u ^{\frac{9}{2}}} - (- \frac{4}{5 u ^{\frac{5}{2}}}) - \frac{2}{u}

Setze in

u = sin (β)

ein

= - \frac{2}{9 sin ^{\frac{9}{2}} ( β )} - (- \frac{4}{5 sin ^{\frac{5}{2}} ( β )}) - \frac{2}{sin ( β )}

Vereinfache

- \frac{2}{9 sin ^{\frac{9}{2}} ( β )} - (- \frac{4}{5 sin ^{\frac{5}{2}} ( β )}) - \frac{2}{sin ( β )} : - \frac{2}{9} csc^{\frac{9}{2}} (β) + \frac{4}{5} csc^{\frac{5}{2}} (β) - 2 csc (β)

= - \frac{2}{9} csc^{\frac{9}{2}} (β) + \frac{4}{5} csc^{\frac{5}{2}} (β) - 2 csc (β)

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{9} csc^{\frac{9}{2}} (β) + \frac{4}{5} csc^{\frac{5}{2}} (β) - 2 csc (β) + C

Graph

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