integral of 1/(4x^2+12x+25)

Lösung

\int \frac{1}{4 x ^{2} + 12 x + 25} d x

\frac{1}{8} arctan (\frac{1}{4} (2 x + 3)) + C

Schritte zur Lösung

\int \frac{1}{4 x ^{2} + 12 x + 25} d x

Vervollständige das Quadrat

4 x^{2} + 12 x + 25 : 4 (x + \frac{3}{2})^{2} + 16

= \int \frac{1}{4 ( x + \frac{3}{2} ) ^{2} + 16} d x

Wende U-Substitution an

= \int \frac{1}{4 u ^{2} + 16} d u

Wende integrale Substitution an

= \int \frac{1}{8 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int \frac{1}{v ^{2} + 1} d v

Nutze das gemeinsame Integral :

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

= \frac{1}{8} arctan (v)

Ersetze zurück

= \frac{1}{8} arctan (\frac{2}{4} (x + \frac{3}{2}))

Vereinfache

\frac{1}{8} arctan (\frac{2}{4} (x + \frac{3}{2})) : \frac{1}{8} arctan (\frac{1}{4} (2 x + 3))

= \frac{1}{8} arctan (\frac{1}{4} (2 x + 3))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} arctan (\frac{1}{4} (2 x + 3)) + C

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