integral of 1/(ax^2+bx+c)

Lösung

\int \frac{1}{a x ^{2} + b x + c} d x

\frac{2}{- b ^{2} + 4 a c} arctan (\frac{2 a x + b}{4 a c - b ^{2}}) + C

Schritte zur Lösung

\int \frac{1}{a x ^{2} + b x + c} d x

Vervollständige das Quadrat

a x^{2} + b x + c : (a x + \frac{b}{2 a})^{2} + c - \frac{b ^{2}}{4 a}

= \int \frac{1}{( a x + \frac{b}{2 a} ) ^{2} + c - \frac{b ^{2}}{4 a}} d x

Wende U-Substitution an

= \int \frac{4 a}{4 a u ^{2} + 4 a c - b ^{2}} d u

Entferne die Konstante:

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

= 4 a \cdot \int \frac{1}{4 a u ^{2} + 4 a c - b ^{2}} d u

Wende integrale Substitution an

= 4 a \cdot \int \frac{1}{2 a ( v ^{2} + 1 ) - b ^{2} + 4 a c} d v

Entferne die Konstante:

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

= 4 a \frac{1}{2 a - b ^{2} + 4 a c} \cdot \int \frac{1}{v ^{2} + 1} d v

Nutze das gemeinsame Integral :

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 4 a \frac{1}{2 a - b ^{2} + 4 a c} arctan (v)

Ersetze zurück

= 4 a \frac{1}{2 a - b ^{2} + 4 a c} arctan (\frac{2 a}{4 a c - b ^{2}} (a x + \frac{b}{2 a}))

Vereinfache

4 a \frac{1}{2 a - b ^{2} + 4 a c} arctan (\frac{2 a}{4 a c - b ^{2}} (a x + \frac{b}{2 a})) : \frac{2}{- b ^{2} + 4 a c} arctan (\frac{2 a x + b}{4 a c - b ^{2}})

= \frac{2}{- b ^{2} + 4 a c} arctan (\frac{2 a x + b}{4 a c - b ^{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{2}{- b ^{2} + 4 a c} arctan (\frac{2 a x + b}{4 a c - b ^{2}}) + C

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