integral of (x^2-3)/(x^3+5x)

Lösung

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

\frac{4}{5} ln x^{2} + 5 - \frac{3}{5} ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} - 3}{x ^{3} + 5 x} : \frac{x ^{2}}{x ^{3} + 5 x} - \frac{3}{x ^{3} + 5 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 ^{2}}{x ^{3} + 5 x} d x - \int \frac{3}{x ^{3} + 5 x} d x

\int \frac{x ^{2}}{x ^{3} + 5 x} d x = \frac{1}{2} ln x^{2} + 5

\int \frac{3}{x ^{3} + 5 x} d x = 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln x^{2} + 5)

= \frac{1}{2} ln x^{2} + 5 - 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln x^{2} + 5)

Vereinfache

\frac{1}{2} ln x^{2} + 5 - 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln x^{2} + 5) : \frac{4}{5} ln x^{2} + 5 - \frac{3}{5} ln ∣ x ∣

= \frac{4}{5} ln x^{2} + 5 - \frac{3}{5} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{5} ln x^{2} + 5 - \frac{3}{5} ln ∣ x ∣ + C

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