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Beliebt Rechnen >

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

Lösung

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

Lösung

x^{2} + x + 1 - \frac{1}{2} ln \frac{2}{3} x + \frac{1}{3} + \frac{4 x ^{2} + 4 x + 4}{3} + C

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

= \int \frac{2 u - 1}{4 u ^{2} + 3} d u

Multipliziere aus

\frac{2 u - 1}{4 u ^{2} + 3} : \frac{2 u}{4 u ^{2} + 3} - \frac{1}{4 u ^{2} + 3}

= \int \frac{2 u}{4 u ^{2} + 3} - \frac{1}{4 u ^{2} + 3} 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

= \int \frac{2 u}{4 u ^{2} + 3} d u - \int \frac{1}{4 u ^{2} + 3} d u

\int \frac{2 u}{4 u ^{2} + 3} d u = \frac{1}{2} 4 u^{2} + 3

\int \frac{1}{4 u ^{2} + 3} d u = \frac{1}{2} ln \frac{2}{3} u + \frac{1}{3} (3 + 4 u^{2})

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

Setze in

u = x + \frac{1}{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= x^{2} + x + 1 - \frac{1}{2} ln \frac{2}{3} x + \frac{1}{3} + \frac{4 x ^{2} + 4 x + 4}{3} + C

Graph

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