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Beliebt Rechnen >

integral of (csc^2(2x))/(cot^5(2x))

Lösung

\int \frac{csc ^{2} ( 2 x )}{cot ^{5} ( 2 x )} d x

Lösung

- \frac{1}{2} sec^{2} (2 x) + \frac{1}{8} sec^{4} (2 x) + \frac{1}{4} sec^{2} (2 x) + C

Schritte zur Lösung

\int \frac{csc ^{2} ( 2 x )}{cot ^{5} ( 2 x )} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1 + cot ^{2} ( 2 x )}{cot ^{5} ( 2 x )} d x

Multipliziere aus

\frac{1 + cot ^{2} ( 2 x )}{cot ^{5} ( 2 x )} : \frac{1}{cot ^{5} ( 2 x )} + \frac{1}{cot ^{3} ( 2 x )}

= \int \frac{1}{cot ^{5} ( 2 x )} + \frac{1}{cot ^{3} ( 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{1}{cot ^{5} ( 2 x )} d x + \int \frac{1}{cot ^{3} ( 2 x )} d x

\int \frac{1}{cot ^{5} ( 2 x )} d x = \frac{1}{2} (ln ∣ sec (2 x) ∣ - sec^{2} (2 x) + \frac{sec ^{4} ( 2 x )}{4})

\int \frac{1}{cot ^{3} ( 2 x )} d x = - \frac{1}{2} ln ∣ sec (2 x) ∣ + \frac{sec ^{2} ( 2 x )}{4}

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

Vereinfache

\frac{1}{2} (ln ∣ sec (2 x) ∣ - sec^{2} (2 x) + \frac{sec ^{4} ( 2 x )}{4}) - \frac{1}{2} ln ∣ sec (2 x) ∣ + \frac{sec ^{2} ( 2 x )}{4} : - \frac{1}{2} sec^{2} (2 x) + \frac{1}{8} sec^{4} (2 x) + \frac{1}{4} sec^{2} (2 x)

= - \frac{1}{2} sec^{2} (2 x) + \frac{1}{8} sec^{4} (2 x) + \frac{1}{4} sec^{2} (2 x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} sec^{2} (2 x) + \frac{1}{8} sec^{4} (2 x) + \frac{1}{4} sec^{2} (2 x) + C

Graph

Beliebte Beispiele

limit as x approaches 1 of (8x+1)/(x+3)

x \to 1 lim (\frac{8 x + 1}{x + 3})

slopeintercept (-3,2),(-1,-1)

s l o p e in t erce pt (- 3, 2), (- 1, - 1)

derivative xsqrt(x^2+9)

d er i v a t i v e x x^{2} + 9

y^{''}+2y^'+10y=0,y(0)=3,y^'(0)=-1

y^{^{''}} + 2 y^{^{'}} + 10 y = 0, y (0) = 3, y^{^{'}} (0) = - 1

derivative of (e^{6x}/x)

\frac{d}{d x} (\frac{e ^{6 x}}{x})