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Beliebt Rechnen >

integral of tan^7(x)sec^{17}(x)

Lösung

\int tan^{7} (x) sec^{17} (x) d x

Lösung

\frac{sec ^{23} ( x )}{23} - \frac{sec ^{21} ( x )}{7} + \frac{3 sec ^{19} ( x )}{19} - \frac{sec ^{17} ( x )}{17} + C

Schritte zur Lösung

\int tan^{7} (x) sec^{17} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (- 1 + sec^{2} (x))^{3} tan (x) sec^{17} (x) d x

Wende U-Substitution an

= \int u^{16} (u^{2} - 1)^{3} d u

Multipliziere aus

u^{16} (u^{2} - 1)^{3} : u^{22} - 3 u^{20} + 3 u^{18} - u^{16}

= \int u^{22} - 3 u^{20} + 3 u^{18} - u^{16} 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^{22} d u - \int 3 u^{20} d u + \int 3 u^{18} d u - \int u^{16} d u

\int u^{22} d u = \frac{u ^{23}}{23}

\int 3 u^{20} d u = \frac{u ^{21}}{7}

\int 3 u^{18} d u = \frac{3 u ^{19}}{19}

\int u^{16} d u = \frac{u ^{17}}{17}

= \frac{u ^{23}}{23} - \frac{u ^{21}}{7} + \frac{3 u ^{19}}{19} - \frac{u ^{17}}{17}

Setze in

u = sec (x)

ein

= \frac{sec ^{23} ( x )}{23} - \frac{sec ^{21} ( x )}{7} + \frac{3 sec ^{19} ( x )}{19} - \frac{sec ^{17} ( x )}{17}

Füge eine Konstante zur Lösung hinzu

= \frac{sec ^{23} ( x )}{23} - \frac{sec ^{21} ( x )}{7} + \frac{3 sec ^{19} ( x )}{19} - \frac{sec ^{17} ( x )}{17} + C

Graph

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