integral of tan^3(x)cos^2(x)

Lösung

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

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ( x )} d x

Wende U-Substitution an

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

Multipliziere aus

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= - ln ∣ u ∣ + \frac{u ^{2}}{2}

Setze in

u = cos (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{x - 2}{x - 3} d x

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