integral of x/(2x^2+3x+2)

Lösung

\int \frac{x}{2 x ^{2} + 3 x + 2} d x

- \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) + \frac{1}{4} ln 16 x^{2} + 24 x + 16 + C

Schritte zur Lösung

\int \frac{x}{2 x ^{2} + 3 x + 2} d x

Wende die partielle Integration an

= \frac{2}{7} x arctan (\frac{1}{7} (4 x + 3)) - \int \frac{2}{7} arctan (\frac{1}{7} (4 x + 3)) d x

\int \frac{2}{7} arctan (\frac{1}{7} (4 x + 3)) d x = \frac{2}{7} x arctan (\frac{1}{7} (4 x + 3)) + \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) - \frac{1}{4} ln 16 x^{2} + 24 x + 16

= \frac{2}{7} x arctan (\frac{1}{7} (4 x + 3)) - (\frac{2}{7} x arctan (\frac{1}{7} (4 x + 3)) + \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) - \frac{1}{4} ln 16 x^{2} + 24 x + 16)

Vereinfache

\frac{2}{7} x arctan (\frac{1}{7} (4 x + 3)) - (\frac{2}{7} x arctan (\frac{1}{7} (4 x + 3)) + \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) - \frac{1}{4} ln 16 x^{2} + 24 x + 16) : - \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) + \frac{1}{4} ln 16 x^{2} + 24 x + 16

= - \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) + \frac{1}{4} ln 16 x^{2} + 24 x + 16

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2 7} arctan (\frac{1}{7} (4 x + 3)) + \frac{1}{4} ln 16 x^{2} + 24 x + 16 + C

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