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Beliebt Rechnen >

integral of (x-1)/((x-1)^2+1)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x}{( x - 1 ) ^{2} + 1} d x = \frac{1}{2} ln x^{2} - 2 x + 2 + arctan (x - 1)

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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