integral of (3x^2+4x+19)/((x+3)(x^2+25))

Lösung

\int \frac{3 x ^{2} + 4 x + 19}{( x + 3 ) ( x ^{2} + 25 )} d x

ln ∣ x + 3 ∣ + ln x^{2} + 25 - \frac{2}{5} arctan (\frac{x}{5}) + C

Schritte zur Lösung

\int \frac{3 x ^{2} + 4 x + 19}{( x + 3 ) ( x ^{2} + 25 )} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} + 4 x + 19}{( x + 3 ) ( x ^{2} + 25 )} : \frac{1}{x + 3} + \frac{2 x - 2}{x ^{2} + 25}

= \int \frac{1}{x + 3} + \frac{2 x - 2}{x ^{2} + 25} 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{1}{x + 3} d x + \int \frac{2 x - 2}{x ^{2} + 25} d x

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

\int \frac{2 x - 2}{x ^{2} + 25} d x = ln x^{2} + 25 - \frac{2}{5} arctan (\frac{x}{5})

= ln ∣ x + 3 ∣ + ln x^{2} + 25 - \frac{2}{5} arctan (\frac{x}{5})

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 3 ∣ + ln x^{2} + 25 - \frac{2}{5} arctan (\frac{x}{5}) + C

\int_{1}^{2} (x^{2} - x) d x

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