integral of (sec(θ))/(sec(θ)-tan(θ))

Lösung

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

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

Schritte zur Lösung

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

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{θ}{2})

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{x ^{2} - 9 x + 26}{x ^{3} + x ^{2} + 4 x + 12} d x

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