integral of 2sqrt(tan(x))sec^4(x)

Lösung

\int 2 tan (x) sec^{4} (x) d x

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

Schritte zur Lösung

\int 2 tan (x) sec^{4} (x) d x

Entferne die Konstante:

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

= 2 \cdot \int tan (x) sec^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

u (1 + u^{2}) : u + u^{\frac{5}{2}}

= 2 \cdot \int u + u^{\frac{5}{2}} 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

= 2 (\int u d u + \int u^{\frac{5}{2}} d u)

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

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

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

Setze in

u = tan (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

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