integral of (tan^8(x))/(sin^5(x))

Lösung

\int \frac{tan ^{8} ( x )}{sin ^{5} ( x )} d x

\frac{1}{35} sec^{5} (x) (- 2 + 5 tan^{2} (x)) + C

Schritte zur Lösung

\int \frac{tan ^{8} ( x )}{sin ^{5} ( x )} d x

Wende U-Substitution an

= \int u^{3} (u^{2} + 1)^{\frac{3}{2}} d u

Wende U-Substitution an

= \int \frac{( v - 1 ) v ^{\frac{3}{2}}}{2} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int (v - 1) v^{\frac{3}{2}} d v

Wende die partielle Integration an

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

\int \frac{2}{5} v^{\frac{5}{2}} d v = \frac{4}{35} v^{\frac{7}{2}}

= \frac{1}{2} (\frac{2}{5} v^{\frac{5}{2}} (v - 1) - \frac{4}{35} v^{\frac{7}{2}})

Ersetze zurück

= \frac{1}{2} (\frac{2}{5} (tan^{2} (x) + 1)^{\frac{5}{2}} (tan^{2} (x) + 1 - 1) - \frac{4}{35} (tan^{2} (x) + 1)^{\frac{7}{2}})

Vereinfache

\frac{1}{2} (\frac{2}{5} (tan^{2} (x) + 1)^{\frac{5}{2}} (tan^{2} (x) + 1 - 1) - \frac{4}{35} (tan^{2} (x) + 1)^{\frac{7}{2}}) : \frac{1}{35} sec^{5} (x) (- 2 + 5 tan^{2} (x))

= \frac{1}{35} sec^{5} (x) (- 2 + 5 tan^{2} (x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{35} sec^{5} (x) (- 2 + 5 tan^{2} (x)) + C

\int \frac{tan ( 4 x )}{cos ( 4 x )} d x

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