integral of 15ln(x^2-x+12)

Lösung

\int 15 ln (x^{2} - x + 12) d x

15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1) + C

Schritte zur Lösung

\int 15 ln (x^{2} - x + 12) d x

Entferne die Konstante:

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

= 15 \cdot \int ln (x^{2} - x + 12) d x

Vervollständige das Quadrat

x^{2} - x + 12 : (x - \frac{1}{2})^{2} + \frac{47}{4}

= 15 \cdot \int ln ((x - \frac{1}{2})^{2} + \frac{47}{4}) d x

Wende U-Substitution an

= 15 \cdot \int ln (u^{2} + \frac{47}{4}) d u

Wende die partielle Integration an

= 15 (u ln (u^{2} + \frac{47}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} + 47} d u)

\int \frac{8 u ^{2}}{4 u ^{2} + 47} d u = - 47 arctan (\frac{2}{47} u) + 2 u

= 15 (u ln (u^{2} + \frac{47}{4}) - (- 47 arctan (\frac{2}{47} u) + 2 u))

Setze in

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

ein

= 15 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{47}{4}) - (- 47 arctan (\frac{2}{47} (x - \frac{1}{2})) + 2 (x - \frac{1}{2})))

Vereinfache

15 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{47}{4}) - (- 47 arctan (\frac{2}{47} (x - \frac{1}{2})) + 2 (x - \frac{1}{2}))) : 15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1)

= 15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1)

Füge eine Konstante zur Lösung hinzu

= 15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1) + C

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