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Beliebt Rechnen >

integral of sqrt(x+\sqrt{x^2+2)}

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{u ^{2} + 2}{u u} : u + \frac{2}{u u}

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

Wende die Summenregel an:

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

= \frac{1}{2} (\int u d u + \int \frac{2}{u u} d u)

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} - \frac{4}{u})

Setze in

u = x + x^{2} + 2

ein

= \frac{1}{2} (\frac{2}{3} (x + x^{2} + 2)^{\frac{3}{2}} - \frac{4}{x + x ^{2} + 2})

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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