integral of sec^4(x)tan^7(x)

Lösung

\int sec^{4} (x) tan^{7} (x) d x

\frac{tan ^{8} ( x )}{8} + \frac{tan ^{10} ( x )}{10} + C

Schritte zur Lösung

\int sec^{4} (x) tan^{7} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

u^{7} (1 + u^{2}) : u^{7} + u^{9}

= \int u^{7} + u^{9} 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

= \int u^{7} d u + \int u^{9} d u

\int u^{7} d u = \frac{u ^{8}}{8}

\int u^{9} d u = \frac{u ^{10}}{10}

= \frac{u ^{8}}{8} + \frac{u ^{10}}{10}

Setze in

u = tan (x)

ein

= \frac{tan ^{8} ( x )}{8} + \frac{tan ^{10} ( x )}{10}

Füge eine Konstante zur Lösung hinzu

= \frac{tan ^{8} ( x )}{8} + \frac{tan ^{10} ( x )}{10} + C

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