integral of tan^{(-3)/2}(x)sec^4(x)

Lösung

\int tan^{\frac{- 3}{2}} (x) sec^{4} (x) d x

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

Schritte zur Lösung

\int tan^{\frac{- 3}{2}} (x) sec^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= \int \frac{1 + u ^{2}}{u ^{\frac{3}{2}}} d u

Multipliziere aus

\frac{1 + u ^{2}}{u ^{\frac{3}{2}}} : \frac{1}{u ^{\frac{3}{2}}} + u^{\frac{1}{2}}

= \int \frac{1}{u ^{\frac{3}{2}}} + u^{\frac{1}{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

= \int \frac{1}{u ^{\frac{3}{2}}} d u + \int u^{\frac{1}{2}} d u

\int \frac{1}{u ^{\frac{3}{2}}} d u = - \frac{2}{u}

\int u^{\frac{1}{2}} d u = \frac{2}{3} u^{\frac{3}{2}}

= - \frac{2}{u} + \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = tan (x)

ein

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

Verwende die grundlegende trigonometrische Identität:

\frac{1}{tan ( x )} = cot (x)

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

Füge eine Konstante zur Lösung hinzu

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

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