integral of (x^2+5x+7)/(x^2+3x+4)

Lösung

\int \frac{x ^{2} + 5 x + 7}{x ^{2} + 3 x + 4} d x

ln x^{2} + 3 x + 4 + x + C

Schritte zur Lösung

\int \frac{x ^{2} + 5 x + 7}{x ^{2} + 3 x + 4} d x

\frac{x ^{2} + 5 x + 7}{x ^{2} + 3 x + 4} = \frac{2 x + 3}{x ^{2} + 3 x + 4} + 1

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

\int \frac{2 x + 3}{x ^{2} + 3 x + 4} d x = ln x^{2} + 3 x + 4

\int 1 d x = x

= ln x^{2} + 3 x + 4 + x

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 3 x + 4 + x + C

\int \frac{x x + 4 x}{x ^{2}} d x

\int 2 - x^{2} + x d x

\int 5 x e^{6 x^{2}} d x

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