de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (sec(x))/(sec(x)-tan(x))

Lösung

\int \frac{sec ( x )}{sec ( x ) - tan ( x )} d x

Lösung

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

Schritte zur Lösung

\int \frac{sec ( x )}{sec ( x ) - tan ( x )} d x

Wende U-Substitution an

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Wende die partielle Integration an

= - 2 (\frac{u}{- u ^{2} + 2 u - 1} - \int \frac{2 u}{( u - 1 ) ^{3}} d u)

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

= - 2 (\frac{u}{- u ^{2} + 2 u - 1} - (- \frac{2}{u - 1} - \frac{1}{( u - 1 ) ^{2}}))

Setze in

u = tan (\frac{x}{2})

ein

= - 2 (\frac{tan ( \frac{x}{2} )}{- tan ^{2} ( \frac{x}{2} ) + 2 tan ( \frac{x}{2} ) - 1} - (- \frac{2}{tan ( \frac{x}{2} ) - 1} - \frac{1}{( tan ( \frac{x}{2} ) - 1 ) ^{2}}))

Vereinfache

- 2 (\frac{tan ( \frac{x}{2} )}{- tan ^{2} ( \frac{x}{2} ) + 2 tan ( \frac{x}{2} ) - 1} - (- \frac{2}{tan ( \frac{x}{2} ) - 1} - \frac{1}{( tan ( \frac{x}{2} ) - 1 ) ^{2}})) : - 2 (\frac{1}{( - 1 + tan ( \frac{x}{2} ) ) ^{2}} + \frac{2}{- 1 + tan ( \frac{x}{2} )} + \frac{tan ( \frac{x}{2} )}{2 tan ( \frac{x}{2} ) - sec ^{2} ( \frac{x}{2} )})

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

limit as x approaches 0-of arcsin(e^x)

x \to 0 - lim (arcsin (e^{x}))

fläche y=6x^2-2x,y=x^3-6x+5

a re a y = 6 x^{2} - 2 x, y = x^{3} - 6 x + 5

derivative of 3x^2(x^3-2x)

\frac{d}{d x} (3 x^{2} (x^{3} - 2 x))

integral from 1 to e^2 of 2/x

\int_{1}^{e^{2}} \frac{2}{x} d x

derivative of 1.58e^{-0.213x}

\frac{d}{d x} (1.58 e^{- 0.213 x})