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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1 - cos ^{2} ( x )}{cos ^{3} ( x )} d x

Multipliziere aus

\frac{1 - cos ^{2} ( x )}{cos ^{3} ( x )} : \frac{1}{cos ^{3} ( x )} - \frac{1}{cos ( x )}

= \int \frac{1}{cos ^{3} ( x )} - \frac{1}{cos ( 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{1}{cos ^{3} ( x )} d x - \int \frac{1}{cos ( x )} d x

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

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

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

Addiere gleiche Elemente:

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

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

Füge eine Konstante zur Lösung hinzu

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Graph

Beliebte Beispiele

fläche y= 26/5-1/x ,y=x

a re a y = \frac{26}{5} - \frac{1}{x}, y = x

fläche 2x^2,2x+6,-2,-1

a re a 2 x^{2}, 2 x + 6, - 2, - 1

limit as x approaches infinity of e^{3x}

x \to \infty lim (e^{3 x})

limit as x approaches 8+of 2/(x-8)

x \to 8 + lim (\frac{2}{x - 8})

taylor (x-1)ln(x-1),2

t a y l or (x - 1) ln (x - 1), 2