integral of (x^2)/(x^2+x+72)

Lösung

\int \frac{x ^{2}}{x ^{2} + x + 72} d x

- \frac{1}{2} ln 4 x^{2} + 4 x + 288 - \frac{143}{287} arctan (\frac{1}{287} (2 x + 1)) + x + C

Schritte zur Lösung

\int \frac{x ^{2}}{x ^{2} + x + 72} d x

\frac{x ^{2}}{x ^{2} + x + 72} = \frac{- x - 72}{x ^{2} + x + 72} + 1

= \int \frac{- x - 72}{x ^{2} + x + 72} + 1 d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{- x - 72}{x ^{2} + x + 72} d x + \int 1 d x

\int \frac{- x - 72}{x ^{2} + x + 72} d x = - \frac{1}{2} ln 4 x^{2} + 4 x + 288 - \frac{143}{287} arctan (\frac{1}{287} (2 x + 1))

\int 1 d x = x

= - \frac{1}{2} ln 4 x^{2} + 4 x + 288 - \frac{143}{287} arctan (\frac{1}{287} (2 x + 1)) + x

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln 4 x^{2} + 4 x + 288 - \frac{143}{287} arctan (\frac{1}{287} (2 x + 1)) + x + C

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