integral of ((sin(x))^2)/((cos(x))^3)

Lösung

\int \frac{( sin ( x ) ) ^{2}}{( cos ( x ) ) ^{3}} d x

\frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Schritte zur Lösung

\int \frac{( sin ( x ) ) ^{2}}{( cos ( x ) ) ^{3}} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

(- 1 + sec^{2} (x)) sec (x) : - sec (x) + sec^{3} (x)

= \int - sec (x) + sec^{3} (x) d x

Wende die Summenregel an:

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

= - \int sec (x) d x + \int sec^{3} (x) d x

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{ln ^{2} ( x )}{x} d x

\int 2 x (x^{2} + 7)^{3} d x

\int 5 x e^{9 x^{2}} d x

\int \frac{x ^{4} + x ^{3} + 6 x + 3}{x ^{3} - x ^{2}} d x

\int \frac{e ^{- x}}{e ^{x} + e ^{- x}} d x