integral of x(3sec^2(x)+2tan^2(x))

Lösung

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

- x^{2} + 5 x tan (x) + 5 ln ∣ cos (x) ∣ + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

\frac{x ( 3 + 2 sin ^{2} ( x ) )}{cos ^{2} ( x )} : \frac{3 x}{cos ^{2} ( x )} + \frac{2 x sin ^{2} ( x )}{cos ^{2} ( x )}

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

\int \frac{3 x}{cos ^{2} ( x )} d x = 6 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣)

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

= 6 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) - x^{2} + 2 x tan (x) + 2 ln ∣ cos (x) ∣

Vereinfache

6 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) - x^{2} + 2 x tan (x) + 2 ln ∣ cos (x) ∣ : - x^{2} + 5 x tan (x) + 5 ln ∣ cos (x) ∣

= - x^{2} + 5 x tan (x) + 5 ln ∣ cos (x) ∣

Füge eine Konstante zur Lösung hinzu

= - x^{2} + 5 x tan (x) + 5 ln ∣ cos (x) ∣ + C

t an g e n t y = 7 4 x^{3}

3 y^{^{''}} + 2 y = x^{2} e^{- 3 x}

\frac{d}{d x} (\frac{x ^{2}}{x ^{2} + x + 1})

t an g e n t y = ln (1 - x^{2} + 2 x^{4}), at x = 1

\frac{d}{d y} (x^{2} + y^{2})