integral of 7sec^6(x)

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= 7 \cdot \int 1 + 2 u^{2} + u^{4} 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

= 7 (\int 1 d u + \int 2 u^{2} d u + \int u^{4} d u)

\int 1 d u = u

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

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

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

Setze in

u = tan (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

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