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integral sqrt(((x-2))/((x-1)))

Lösung

integral

\frac{( x - 2 )}{( x - 1 )}

x - 2 x - 1 - ln x - 2 + x - 1 + C

Schritte zur Lösung

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

Vereinfache

\frac{x - 2}{x - 1} : \frac{x - 2}{x - 1}

= \int \frac{x - 2}{x - 1} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Trigonometrische Substitution anwenden

= 2 \cdot \int tan^{2} (v) sec (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

Multipliziere aus

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 2 \cdot \int - sec (v) + sec^{3} (v) d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (- \int sec (v) d v + \int sec^{3} (v) d v)

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

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

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

Ersetze zurück

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

Vereinfache

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

= x - 2 x - 1 - ln x - 2 + x - 1

Füge eine Konstante zur Lösung hinzu

= x - 2 x - 1 - ln x - 2 + x - 1 + C

in t e g r a l ln ((\frac{1}{x}) - 1)

in t e g r a l \frac{( x ^{2} - a ) ^{3}}{2}

in t e g r a l t (1 - t)

in t e g r a l \frac{x}{( x ^{2} + 1 )}

in t e g r a l x^{2} (sin (x))^{2}