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Beliebt Rechnen >

integral of ln(x^2+x+2)

Lösung

\int ln (x^{2} + x + 2) d x

Lösung

x ln (x^{2} + x + 2) + \frac{1}{2} ln (x^{2} + x + 2) + 7 arctan (\frac{1}{7} (2 x + 1)) - 2 x - 1 + C

Schritte zur Lösung

\int ln (x^{2} + x + 2) d x

Vervollständige das Quadrat

x^{2} + x + 2 : (x + \frac{1}{2})^{2} + \frac{7}{4}

= \int ln ((x + \frac{1}{2})^{2} + \frac{7}{4}) d x

Wende U-Substitution an

= \int ln (u^{2} + \frac{7}{4}) d u

Wende die partielle Integration an

= u ln (u^{2} + \frac{7}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} + 7} d u

\int \frac{8 u ^{2}}{4 u ^{2} + 7} d u = - 7 arctan (\frac{2}{7} u) + 2 u

= u ln (u^{2} + \frac{7}{4}) - (- 7 arctan (\frac{2}{7} u) + 2 u)

Setze in

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

ein

= (x + \frac{1}{2}) ln ((x + \frac{1}{2})^{2} + \frac{7}{4}) - (- 7 arctan (\frac{2}{7} (x + \frac{1}{2})) + 2 (x + \frac{1}{2}))

Vereinfache

(x + \frac{1}{2}) ln ((x + \frac{1}{2})^{2} + \frac{7}{4}) - (- 7 arctan (\frac{2}{7} (x + \frac{1}{2})) + 2 (x + \frac{1}{2})) : x ln (x^{2} + x + 2) + \frac{1}{2} ln (x^{2} + x + 2) + 7 arctan (\frac{1}{7} (2 x + 1)) - 2 x - 1

= x ln (x^{2} + x + 2) + \frac{1}{2} ln (x^{2} + x + 2) + 7 arctan (\frac{1}{7} (2 x + 1)) - 2 x - 1

Füge eine Konstante zur Lösung hinzu

= x ln (x^{2} + x + 2) + \frac{1}{2} ln (x^{2} + x + 2) + 7 arctan (\frac{1}{7} (2 x + 1)) - 2 x - 1 + C

Graph

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