integral of sec^6(7x)

Lösung

\int sec^{6} (7 x) d x

\frac{1}{7} (tan (7 x) + \frac{2 tan ^{3} ( 7 x )}{3} + \frac{tan ^{5} ( 7 x )}{5}) + C

Schritte zur Lösung

\int sec^{6} (7 x) d x

Wende U-Substitution an

= \int sec^{6} (u) \frac{1}{7} d u

Entferne die Konstante:

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

= \frac{1}{7} \cdot \int sec^{6} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{7} \cdot \int sec^{2} (u) (1 + tan^{2} (u))^{2} d u

Wende U-Substitution an

= \frac{1}{7} \cdot \int (1 + v^{2})^{2} d v

Multipliziere aus

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

= \frac{1}{7} \cdot \int 1 + 2 v^{2} + v^{4} 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}{7} (\int 1 d v + \int 2 v^{2} d v + \int v^{4} d v)

\int 1 d v = v

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

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

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

Ersetze zurück

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} (tan (7 x) + \frac{2 tan ^{3} ( 7 x )}{3} + \frac{tan ^{5} ( 7 x )}{5}) + C

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