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Beliebt Rechnen >

integral of 1/(sqrt(x^2-2x-3))

Lösung

\int \frac{1}{x ^{2} - 2 x - 3} d x

Lösung

ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1) + C

Schritte zur Lösung

\int \frac{1}{x ^{2} - 2 x - 3} d x

Vervollständige das Quadrat

x^{2} - 2 x - 3 : (x - 1)^{2} - 4

= \int \frac{1}{( x - 1 ) ^{2} - 4} d x

Wende U-Substitution an

= \int \frac{1}{u ^{2} - 4} d u

Trigonometrische Substitution anwenden

= \int sec (v) d v

Nutze das gemeinsame Integral :

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

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

Vereinfache

ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1))) : ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1)

= ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1)

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x/(x^2-5x+3)

\int \frac{x}{x ^{2} - 5 x + 3} d x

integral of (2x^2)/(x^2+3)

\int \frac{2 x ^{2}}{x ^{2} + 3} d x

integral of (2x+7)(x^2+7x+1)

\int (2 x + 7) (x^{2} + 7 x + 1) d x

integral of ((6x-5))/(sqrt(3x^2-5x+6))

\int \frac{( 6 x - 5 )}{3 x ^{2} - 5 x + 6} d x

integral of ((x^3)/3+2)^2x^3

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