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integral of-5x^2sec^2(3x^3)tan^5(3x^3)

Lösung

\int - 5 x^{2} sec^{2} (3 x^{3}) tan^{5} (3 x^{3}) d x

- \frac{5}{54} (sec^{6} (3 x^{3}) - 3 sec^{4} (3 x^{3}) + 3 sec^{2} (3 x^{3})) + C

Schritte zur Lösung

\int - 5 x^{2} sec^{2} (3 x^{3}) tan^{5} (3 x^{3}) d x

Entferne die Konstante:

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

= - 5 \cdot \int x^{2} sec^{2} (3 x^{3}) tan^{5} (3 x^{3}) d x

Wende U-Substitution an

= - 5 \cdot \int \frac{tan ^{5} ( 3 u ) sec ^{2} ( 3 u )}{3} d u

Entferne die Konstante:

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

= - 5 \cdot \frac{1}{3} \cdot \int tan^{5} (3 u) sec^{2} (3 u) d u

Wende U-Substitution an

= - 5 \cdot \frac{1}{3} \cdot \int tan^{5} (v) sec^{2} (v) \frac{1}{3} d v

Entferne die Konstante:

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

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \int tan^{5} (v) sec^{2} (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \int (- 1 + sec^{2} (v))^{2} tan (v) sec^{2} (v) d v

Wende U-Substitution an

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \int w (w^{2} - 1)^{2} d w

Multipliziere aus

w (w^{2} - 1)^{2} : w^{5} - 2 w^{3} + w

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \int w^{5} - 2 w^{3} + w d w

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} (\int w^{5} d w - \int 2 w^{3} d w + \int w d w)

\int w^{5} d w = \frac{w ^{6}}{6}

\int 2 w^{3} d w = \frac{w ^{4}}{2}

\int w d w = \frac{w ^{2}}{2}

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} (\frac{w ^{6}}{6} - \frac{w ^{4}}{2} + \frac{w ^{2}}{2})

Ersetze zurück

= - 5 \cdot \frac{1}{3} \cdot \frac{1}{3} (\frac{sec ^{6} ( 3 x ^{3} )}{6} - \frac{sec ^{4} ( 3 x ^{3} )}{2} + \frac{sec ^{2} ( 3 x ^{3} )}{2})

Vereinfache

- 5 \cdot \frac{1}{3} \cdot \frac{1}{3} (\frac{sec ^{6} ( 3 x ^{3} )}{6} - \frac{sec ^{4} ( 3 x ^{3} )}{2} + \frac{sec ^{2} ( 3 x ^{3} )}{2}) : - \frac{5}{54} (sec^{6} (3 x^{3}) - 3 sec^{4} (3 x^{3}) + 3 sec^{2} (3 x^{3}))

= - \frac{5}{54} (sec^{6} (3 x^{3}) - 3 sec^{4} (3 x^{3}) + 3 sec^{2} (3 x^{3}))

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{54} (sec^{6} (3 x^{3}) - 3 sec^{4} (3 x^{3}) + 3 sec^{2} (3 x^{3})) + C

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