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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

\int \frac{1}{x ^{2} + 2 x + 2} d x = ln x^{2} + 2 x + 2 + x + 1

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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