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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} + 4 x - 5}{x ^{3}} : \frac{1}{x} + \frac{4}{x ^{2}} - \frac{5}{x ^{3}}

= \int \frac{1}{x} + \frac{4}{x ^{2}} - \frac{5}{x ^{3}} 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} d x + \int \frac{4}{x ^{2}} d x - \int \frac{5}{x ^{3}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{4}{x ^{2}} d x = - \frac{4}{x}

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

= ln ∣ x ∣ - \frac{4}{x} - (- \frac{5}{2 x ^{2}})

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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