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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere mit der Konjugation von

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

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

Entferne die Konstante:

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

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

Vereinfache

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

\int x d x = \frac{x ^{2}}{2}

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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