integral of (sqrt(cot(β)))/(sin^6(β))

Lösung

\int \frac{cot ( β )}{sin ^{6} ( β )} d β

- cot (β) cot (β) - \frac{2}{3} cot^{3} (β) cot (β) - \frac{1}{5} cot^{5} (β) cot (β) + \frac{1}{55} cot^{\frac{11}{2}} (β) + \frac{2}{21} cot^{\frac{7}{2}} (β) + \frac{1}{3} cot^{\frac{3}{2}} (β) + C

Schritte zur Lösung

\int \frac{cot ( β )}{sin ^{6} ( β )} d β

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

= - (u (u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}) - \int \frac{1}{30} u (15 + 10 u^{2} + 3 u^{4}) d u)

\int \frac{1}{30} u (15 + 10 u^{2} + 3 u^{4}) d u = \frac{1}{1155} u^{\frac{3}{2}} (21 u^{4} + 110 u^{2} + 385)

= - (u (u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}) - \frac{1}{1155} u^{\frac{3}{2}} (21 u^{4} + 110 u^{2} + 385))

Setze in

u = cot (β)

ein

= - (cot (β) (cot (β) + \frac{2 cot ^{3} ( β )}{3} + \frac{cot ^{5} ( β )}{5}) - \frac{1}{1155} cot^{\frac{3}{2}} (β) (21 cot^{4} (β) + 110 cot^{2} (β) + 385))

Vereinfache

- (cot (β) (cot (β) + \frac{2 cot ^{3} ( β )}{3} + \frac{cot ^{5} ( β )}{5}) - \frac{1}{1155} cot^{\frac{3}{2}} (β) (21 cot^{4} (β) + 110 cot^{2} (β) + 385)) : - cot (β) cot (β) - \frac{2}{3} cot^{3} (β) cot (β) - \frac{1}{5} cot^{5} (β) cot (β) + \frac{1}{55} cot^{\frac{11}{2}} (β) + \frac{2}{21} cot^{\frac{7}{2}} (β) + \frac{1}{3} cot^{\frac{3}{2}} (β)

= - cot (β) cot (β) - \frac{2}{3} cot^{3} (β) cot (β) - \frac{1}{5} cot^{5} (β) cot (β) + \frac{1}{55} cot^{\frac{11}{2}} (β) + \frac{2}{21} cot^{\frac{7}{2}} (β) + \frac{1}{3} cot^{\frac{3}{2}} (β)

Füge eine Konstante zur Lösung hinzu

= - cot (β) cot (β) - \frac{2}{3} cot^{3} (β) cot (β) - \frac{1}{5} cot^{5} (β) cot (β) + \frac{1}{55} cot^{\frac{11}{2}} (β) + \frac{2}{21} cot^{\frac{7}{2}} (β) + \frac{1}{3} cot^{\frac{3}{2}} (β) + C

\int \frac{1}{1 + \frac{1}{x}} d x

\frac{d}{d x} (4 x + 2)

\frac{d}{d x} (\frac{8}{3 + x ^{2}})

\int x^{\frac{1}{2}} d x

\frac{d}{d x} (\frac{y - 2 x}{2 y - x})