integral of x(2sec^2(x)+tan^2(x))

Lösung

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

- x^{2} + \frac{x ^{2}}{2} + 3 x tan (x) + 3 ln ∣ cos (x) ∣ + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

\frac{x ( 2 + sin ^{2} ( x ) )}{cos ^{2} ( x )} : \frac{2 x}{cos ^{2} ( x )} + \frac{x sin ^{2} ( x )}{cos ^{2} ( x )}

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

\int \frac{2 x}{cos ^{2} ( x )} d x = 4 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣)

\int \frac{x sin ^{2} ( x )}{cos ^{2} ( x )} d x = - x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣

= 4 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) - x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣

Vereinfache

4 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) - x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣ : - x^{2} + \frac{x ^{2}}{2} + 3 x tan (x) + 3 ln ∣ cos (x) ∣

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

Füge eine Konstante zur Lösung hinzu

= - x^{2} + \frac{x ^{2}}{2} + 3 x tan (x) + 3 ln ∣ cos (x) ∣ + C

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