integral of 8sec^6(t)

Lösung

\int 8 sec^{6} (t) d t

8 (tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5}) + C

Schritte zur Lösung

\int 8 sec^{6} (t) d t

Entferne die Konstante:

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

= 8 \cdot \int sec^{6} (t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 8 \cdot \int sec^{2} (t) (1 + tan^{2} (t))^{2} d t

Wende U-Substitution an

= 8 \cdot \int (1 + u^{2})^{2} d u

Multipliziere aus

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

= 8 \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

= 8 (\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}

= 8 (u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5})

Setze in

u = tan (t)

ein

= 8 (tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5})

Füge eine Konstante zur Lösung hinzu

= 8 (tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5}) + C

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