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

Lösung

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

\frac{1}{2} cos^{2} (x) + ln ∣ sec (x) ∣ + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

= - (- \frac{1}{2 u ^{2}}) + ln ∣ u ∣

Setze in

u = sec (x)

ein

= - (- \frac{1}{2 sec ^{2} ( x )}) + ln ∣ sec (x) ∣

Vereinfache

- (- \frac{1}{2 sec ^{2} ( x )}) + ln ∣ sec (x) ∣ : \frac{1}{2} cos^{2} (x) + ln ∣ sec (x) ∣

= \frac{1}{2} cos^{2} (x) + ln ∣ sec (x) ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} cos^{2} (x) + ln ∣ sec (x) ∣ + C

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